From a strictly football perspective, it makes sense. It's not a major plus. It's a low-risk type of acquisition. As a #3 QB you have a guy with NFL experience, with athleticism, with playoff experience. His mechanics suck, his decision making is questionable, but as a #3 last resort, project QB, why not? What is there to lose?
And he is an athlete and can play in different spots, maybe as a goal-line runningback, maybe as a #4 tight-end, whatever.
All his on-field negatives as a passer are irrelevant because I don't think he'll see the field as a quarterback unless something awful has happened or the Patriots are up by 28 points with a minute left. And in other spots he'll see the field if he deserves it. Hopefully not in many trick situations, Josh McDaniels likes those too much in my opinion.
The Patriots drafted a rugby player once. They signed a track athlete. They're willing to give athletes a chance to earn a spot on the roster and on the field. They now have a #3 QB who might be able to find minor roles in other aspects of the game. Why not?
As far as the off-field negatives (in this day and age we consider someone with no criminal record to have off-field negatives), the whole "circus" that accompanies Tebow, I'm not too worried. There won't be any QB controversy between he and Brady. The media interns at Gillette Stadium will have to print more credentials for training camp. And the Patriots coaches and players will have more questions to answer without really answering. What team is better at defeating the media's attempts to excavate interesting stories and extract juicy quotes than the Patriots?
Bill Belichick vs. The Media. Who's the favorite in that battle?
This is a huge story for a #3 quarterback who might play special-teams and goal-line situations. This is a potential minor improvement to the Patriots roster with a player who has garnered far too much attention, praise, and hatred for who and what he has done. The Tim Tebow story is a feedback loop. The media is making a circus of it because it is a media circus. This is a story about a story.
If you're like me and emotionally upset about this, I'm going to quote The Book of Rob:
"This isn't the end of the world."
-Rob 3:16
