So why would Belichick sign Tebow?
Because he can. It really might not be more complicated than that. Former Denver Broncos coach Josh McDaniels remains on the Patriots' staff and obviously is a big believer in Tebow. There virtually is no risk to the signing because Tebow's contract is unlikely to have any guarantees. The Patriots love to kick the tires on veterans, and this is nothing more than that.
Tebow, like a lot of No. 3 quarterbacks in the NFL, must prove his worthiness in limited practice snaps. Unlike most No. 3 quarterbacks, however, he could help on special teams and the occasional trick play.