I don't think he had any intentions of keeping Stidham even for free.I've had to remind myself something and it helped make sense of this draft. Otherwise it doesn't make sense or can be extremely frustrating.
Belichick views drafts like a ruthless economist.
So, picks will be done to save money or get efficient with some positions cap-wise, not necessarily to improve talent at a key position.
In this case with the QB, it's about not extending Stidham later. It's about saving money. You keep a cheap backup QB on the roster and don't pay Stidham low millions per year to stay on later. Also explains the RB pick. The Pats already are strong at the RB position. But maybe they don't want to spend multimillions coming up to keep a RB post-rookie deal. So they use a revolving group of cheap drafted RB guys and don't commit $ to extending RBs.
Having said all that, it sucks to then waste precious cap on expensive crappy players in Agholor, Godchaux, Jonnu, and Wynn at 11-14M each. Or to not keep JC Jackson or Mason who are actually elite players. Or to not have fixed the terrible run defense at all. Right now the organizational actions are not completely congruent.