That might be a bit of an over-simplification...
In the 2-hour "signing exercise" you had to go through when you closed on your house, how sure are you that you didn't sign/initial a clause that authorizes your first lender to sell or otherwise transfer your mortgage to another holder? That has been a standard clause in each of the three mortgages I have had in my lifetime (so far) over the past 25 years. That's with three different lenders.
I'm not sure how those clauses would affect the use-cases offered as examples so far, but I would think it would show you had *some* liability to repay the loan in the event it was transferred/sold.
However, if it is the case where the original documents are well and truly destroyed, eliminating any and all instances bearing your signature/initials, then I can see where you would have an argument of "prove I signed it." But, if you are the home-owner of record (property tax rolls), I would hope for a very tolerant judge before I tried that argument.
Of course, I will readily admit I'm not a SME. I'm just trying to apply a bit of common sense to the issue.
Bank 'A' would not mark your note as paid in full, they would assign the mortgage, and endorse the Note over to the new Investor/servicer.
Please excuse my impatience with this thread as I have 25 years in the mortgage banking business and thost that think borrowers have no obligation to repay because thier loas were sold are sadly misinformed.