Now that Tim Tebow is a free agent, with (at time of writing) no team willing to sign him to any playing contract, it is time to wonder out loud: how did we get here?
How did one of the premier impact players of this current generation in the college game, drafted in the first round of the NFL draft, end up 3 years later almost consigned to the scrap-heap?
In reading all of the millions of bits and bytes written on this subject, and having watched Tebow in the NFL, here is my analysis of the possible answers to Why?

1. The NFL is a system-driven league
There are a couple of basic realities about the NFL that become obvious to watchers after a while. Firstly, the NFL coaching process is dominated by "systems". What is known as "The West Coast" offense is a system first used by Bill Walsh in San Francisco and now used in a number of NFL franchises. The overall approach is derived from the 49'ers, and the terminology is often similar between teams. I have lost count of the number of times that I have read that a player being talked about (usually a free agent or a soon-to-be free agent) would be a good fit for a team because "he knows the system". In fact, there is a lot of evidence that some teams, given the choice between a moderately talented player who is perceived to know the system, and a more talented player who does not know the system, will opt to sign the "system" player, on the grounds that the player can more quickly and easily fit into their team, be it offense, defense or special teams. Interestingly, the New England Patriots do not seem to suffer from this false dichotomy worldview. Their actions in signing players over the years show that they are only interested in the best players, and they will then work out how to use them most effectively.
Secondly, most NFL teams are deeply conservative in their offensive schemes, play-calling and general conduct of offenses. Gregg Easterbrook has pointed out many times with examples, how coaches, when their team is behind in a game, often prefer to play to keep the margin of losing down, rather than to go for the win, or at least try to win.
Many head coaches are also leery of a quarterback with a track record of improvising his way through a game. They expect that the quarterback, lke a good soldier, will always run the play radioed in from the sideline. A handful of quarterbacks (Peyton Manning being the best example) do have multiple plays to choose from at the line of scrimmage, and there are allowed "audibles" for many QBs if they see a certain alignment in the defense after breaking the huddle, but a lot of head coaches still expect the quarterback to run the play as called in from the sideline. A quote from Mike Holmgren from some years back is typical. "If you are going to change one of my plays at the line of scrimmage, it had better work", he once said, referring at the time to Matt Hasselbeck. Having seen Hasselbeck get chewed out by Holmgren when he changed a play and it did not work, I can see why quarterbacks end up doing as Coach wants.

2. Tim Tebow is a natural leader
One thing on which most commentators are agreed, no matter what their basic opinions are of his abilities as a quarterback - Tim Tebow, as a football player, has the charisma and confidence of a natural leader. He is also, like true leaders, prepared to go out and put his body on the line to try and win plays and games. Players respect that level of courage and committment.
Commentators have noted how, while Mark Sanchez was alone in the Jets locker room, Tebow was surrounded by team mates. If you believe in the definition that you can recognize a leader by the presence of willing followers, then Tim Tebow scores more highly than most other players in the league.

3. Tim Tebow is a "project" as a quarterback
Despite having played quarterback since high school (although he began in high school playing as a tight end), Tebow is reckoned by most observers to be far from the finished article as a quarterback. Indeed, he has been the recipient of a massive collection of sometimes contradictory observations, inputs and advice from all manner of ex-players, anonymous coaches, GMs and other opinionators. The criticism ranges from his lack of speed (he is big and powerful, but not reckoned to be agile or fast), through this throwing mechanics, which are supposed to give him a long release, to his footwork, to an inability to read NFL defenses (at which point some commentators fixate on his home-schooling, hinting at that as a cause of poor intellectual development).
The accumulated pile of criticism, some of it public and withering, would be enough to cause most normal mortals to run and hide. To his credit, Tebow has kept quiet about the criticism, and, unlike some recent NFL quarterbacks, he has enlisted the help of retired quarterbacks to try and improve his throwing mechanics and footwork. The extent to which they have been successful will only become apparent over time. Players in stress situations in games tend to revert to old habits.

After the Denver Broncos drafted Tebow in 2010, they did the right thing when they hit a slump in 2011 and decided to try using him. They then went all in, even to the extent of releasing previous starter Kyle Orton, who ended up signing with Dallas. They tailored their offense for Tebow, simpifying it and making it more like a college option-read offense. It worked to the extent that the Broncos not only made the post-season, but beat the Steelers. However, they then lost heavily to the Patriots.
Nevertheless, the Broncos elected to go with Peyton Manning in 2012 upon deciding that he was a better medium-term option. They also traded Tebow out of town instead of keeping him as a backup, and went all in with Peyton Manning, giving him a lot of his old players as targets, and adopting a lot of the Indianapolis Colts offensive system more or less unaltered. They made the post-season, but lost in the first playoff game.
The Broncos probably do not want to be reminded of this, but it is a fact - Tim Tebow has won more playoff games for the Broncos than Peyton Manning thus far.
The New York Jets, having acquired Tebow after the Broncos put him up for trade, proceeded to, let's not be too polite about it, completely eff up their acquisition. For reasons that have never been made clear, Tebow did not gel with the coaching staff, specifically offensive co-ordinator Tony Sparano and head coach Rex Ryan. He was used sparingly in games, and suffered rib injuries, meaning that he was not available when Mark Sanchez was benched for inconsistent play. Rex Ryan, instead of being able to call on the guy who had already won an NFL playoff game, had to play Greg McElroy, who had won no NFL regular season games whatsoever. Not surprisingly, Tebow was intensely frustrated.
Thereafter, Tim Tebow became the invisible man, seen wandering the touchline in games, helmet in hand, ready to go on the field, but rarely called, and when he was called, it was always for some oddball play (the Wildcat having seemingly been ruled out, despite Sparano's introduction of it while he was the head coach of the Dophins) that rarely yielded a positive result. In summary, after a short period of time, the Jets behaved like a team with no clue how to use him, and no interest in using him.
Having whittled down what little trade value Tebow had by reducing him to The Invisible Man on the field, the Jets still squatted on him through the first part of the 2013 off-season, only finally releasing him once it became clear that they would never get any trade offer for him, not even on draft weekend. In truth, no team was going to make any sort of trade offer, once the GM that traded for him was fired, and his replacement hinted that Tebow, as the last bright idea of the previous personnel leadership, was Going To Leave Town Real Soon.
The approach by the Jets resembles the attitude adopted by the Tampa Bay Buccaneers with Chris Simms, where the Bucaneers squatted on Simms all through an offseason after he had nearly died of blood loss when suffering a ruptured spleen, to finally release him at a point in the year when he was not able to find an opportunity of any usefulness with another team. That mean-spirited move more or less ended Simms' career.

So, why do NFL teams behave like Tim Tebow is radioactive? There are four reasons, three of which I will group together.

1. He is a "project" if you want a system quarterback
2. He plays better than he practices
There are really two ways to try and use Tim Tebow in the NFL. One is as a standard backup, running the same playbook. The problem with that approach is that very few teams think he can run a full standard NFL playbook, and they don't want to take snaps from the #1 quarterback in pre-season or in actual game situations to find out.
Second option for using him is as a "wild card" player in a different offense, most commonly in short-yardage situations. This plays to his strengths as a runner and blocker, but the downside is that most of the time he will be running with the ball, not throwing it. That usage pattern will not allow him to throw the ball around the field in a game situation, so it will most likely not allow him to work on accuracy.
I don't know where I read it, but one of the comments out of Denver when Tebow and Kyle Orton were on the roster was that Orton practised better than he played, whilst Tebow was the other way round. In other words, the coaching staff would be wincing as Tebow stunk up the joint during the week, only to exhale (eventually) when he did something freakishly effctive on a Sunday.

3. If he is a backup, the #1 quarterback will be terrified unless he is also a proven leader
4. If he is a backup, every time the team falls behind, the coaches will feel the "put Tebow in" vibe filling their minds and headsets, even if that is not the right answer
5. Tim Tebow = unpredictability
The big problem (ironic that it is now a problem) is that Tebow behaves and operates like a leader all the time. He commands attention and respect, and players instinctively look up to him. This creates tension between the natural caution of coaches (who want a player who knows his role in the system, and predictable outcomes) and players, many of whom are motivated by being around obvious leaders. Quarterbacks like Peyton Manning, Tom Brady and Drew Brees command the confidence of both coaches and players. They have the track records, and they can run all of the aspects of an offense. When they march out on the field, their teams feel good about what is planned most of the time.
When Tim Tebow marches out on the field, three things are true:

1. the coaches have no real idea what will happen. this worries them
2. the players are not sure what will happen, but they believe that Tebow may well pull something out of a hat
3. the spectators expect that Tebow will once again surprise everybody (be it amazing, good, bad or ugly)

In short, Tebow brings unpredictability by the truckload. Given the notorious planning and attention to detail that many NFL coaching staffs obsess over all year round, that level of unpredictability scares coaches and team leaderships witless.
The conservatism of many NFL coaches also works against "wild card" approaches. If a team is behind in the second half of a game, typically they try to recover by throwing the ball down the field - an approach doomed to failure if your quarterback is inaccurate. Trying to advance via trick plays or quarterback improvisation is not often employed.
When coaches hear the "Tebow! Tebow! Tebow!" chant, they will be in a bind. Every neuron of their logical mind is protesting "suicide!".

All is not lost, however.
There are handful of teams that could take Tebow as a backup. If you assume that the right fit is a team with an established #1 quarterback who will not be threatened by Tebow's arrival, then there are handful of candidates.
Broncos, Patriots, Seahawks, Saints, 49ers, Giants, Green Bay Packers, Baltimore Ravens.
All of the other teams have quarterbacks who, to greater or lesser degrees are lower tier (but established) or where the phrases "the jury is out" or "eek, what will happen on this play?" spring to mind whenever they march onto the field with their offense.
Rule out the Broncos, for obvious reasons.
The Patriots have a good backup in Ryan Mallett, who they might trade if they get a great offer, but who is set to remain there for now. Tom Brady is a prototypical pocket passer. The Patriots would be capable of creating plays for Tebow, and offensive coach Josh McDaniel did draft him for the Broncos, but it is far from certain that the Patriots want to make the effort.
The Seahawks do need a mobile quarterback to back up Russell Wilson. They traded away his primary backup, Matt Flynn, to the Raiders.
The Saints have a top-class incumbent in Drew Brees, and not much depth behind him. They do, however, run a fairly conventional pocket-passing offense. But..Sean Payton is one of the more daring offensive coaches in the NFL. He would certainly be able to devise plays to use Tebow effectively.
The 49ers, having traded away Alex Smith, might be interested. They will be using zone read as a key part of the offense with Colin Kaepernick.
The Giants have David Carr to back up Eli Manning, and they run a very conventional pocket-passing offense.
The Packers have Aaron Rodgers, with his new long-term contract, under center. Their offense is primarily pocket-based, but Mike McCarthy seems to be creative, and could probably work out how to use Tebow.
The Ravens have the latest very-rich Superbowl MVP under center, again a primarily pocket-passing attack (and Flacco has shown that he can throw a very accurate deep ball). Superficially, not a likely landing spot.
All of the remaining teams would probably be terrified of destabilizing their incumbent quaterback by signing Tim Tebow. Imagine Tebow marching into the locker rooms at the Browns, Dolphins or Bills. He would immediately have the incumbent QB of the week or month worried, and this would soon translate into all sorts of bad events. Indeed, as I write this, the Dolphins have dropped the strongest possible hint that they have no intention of signing Tebow, despite their need to improve ticket sales.
Where else? The CFL is an obvious option, with the Montreal Alouettes holding Tebow's CFL rights. However, the CFL runs three downs only instead of four, which makes throwing the ball the dominant method of advancing down the field. That does not seem like the best fit for a quarterback whose accuracy is suspect. Right now, Tebow in the CFL would be learning how to throw behind a starter, for less money than in the NFL. However, he would be out of the spotlight, relatively speaking. He would also be available for the last part of the regular NFL season, if a team suddenly loses a quarterback, or finds it needs a "wild card" player.
Move to another position? The problem with that idea is that, as his Wikipedia bio shows, the only other position that Tim Tebow has actually played (as opposed to impersonating during "gadget" plays) is tight end, which was his first high school football position. He soon moved to quarterback, where he has played through high school, college and the NFL. He has never played wide receiver, tailback or running back, so optimists thinking he can move to any of those positions may be truly whistling in the wind. At this stage in his career, switching positions is fraught with danger.

My verdict? The CFL, one of a small handful of NFL teams (both scenarios require him to accept a backup role) or retirement from professional football.
Crazy? Yes.

UPDATE - Following a couple of interesting articles that went beyond the conventional "he sucks" meme to examine some underlying realities (one them being that Tebow, as a highly kinesthetic learner, has trouble with conventional NFL playbooks and instruction modes), the news has broken that he has been signed...by the New England Patriots, one of a handful of teams who can definitely manage Tebow's media interactions. The Patriots have always ensured that their players talk as little as possible to the media, and when they talk, that they reveal as little as possible. Watch for Tim Tebow to have a very low profile indeed in the next few months...