You are missing the point that the bank does not own the house, the owner does despite the loan being in default.
So, the bank has to sell the property to realise the funds and give any profit back to the owner. In the US most loans are non-recourse so the borrower is not required to pay any shortfall.
However, there may be some quirk in American law that means your idea could work. The above is the Australian position, apart from the non-recourse comment.
