If a homeowner can pay their mortgage but refuses to pay the bank—and insists on paying only the bank’s funding source—what happens legally and in practice?

Not sure who told you this, but even int he court of law if you go in for a dispute, the cost of funds to the borrowing bank (from which you took the loan/mortgage) would still be due. Your mortgage is a consent that you will pay the amount, and as such, the cost of funds to the borrowing bank is the very minimum you will have to pay if a dispute does go into your favor by a banking court, etc.