Someone living in that house, however, who doesn't want to move, gets a lot more value out of it than the $20k it's worth to other people, and would therefore be willing to make payments to stay in that house that are higher than what you could get if you took the house from them and resold it.

i doubt that's true in the general case; otherwise nobody would ever foreclose on a mortgage. they would always cut a deal to keep the payments trickling in.