At the risk of having my comment deleted, there was never any proof that he did destroy mortgages.

“There may have been at least a kernel of truth to the legend, based on his reputation for generosity to the “hill people” who helped him hide from the law in the foothills of the Ozarks, according to Jeffery S. King’s book, The Life and Death of Pretty Boy Floyd. King claims Floyd was paying to feed a dozen families who might have otherwise gone hungry”

However, King himself notes again that this was more anecdotal, with no one able to prove it. Woody Guthrie’s song lionizing him also helped spread the rumor.

Michael Wallis notes in his work that bank mortgages even in the thirties had multiple copies, notary offices, city tax and record departments, even other banks would have them. In the 30’s, the bank would drop the notarized copy of the mortgage to sheriffs office, who in turn run it by the mortgage records the local government had, if repossession was on the agenda for nonpayment.

Sources

Pretty Boy: The Life and Times of Charles Arthur Floyd Michael Wallis

The Life and Death of Pretty Boy Floyd Jeffery S. King

Oddly, I can answer this in my professional capacity rather than in my academic capacity.

~40 hours a week I'm a lawyer employed by a state government agency. I also teach a variety of college level and law classes as an adjunct professor.

First, note the other response which said there's actually very little evidence he destroyed mortgages and the top level answer is that even if he did, it was unlikely it would have had as much effect as is popularly suggested. I've read Jeffrey King's book on Pretty Boy Floyd and would absolutely recommend it.

This answer builds off of that answer, because this explains why it wouldn't have much effect.

Legally, when you execute a mortgage, two different documents are involved. There is the "note," which is the legal contract that obligates you to make a certain number of payments of a certain amount to the bank or the holder of the note. Then there is the "Mortgage," which is a document reflecting that the Bank has a legal interest in the property you borrowed money on. It places a lien on the property, so if the property is sold, the bank has a right to collect money from the sale before anyone else.

When the bank creates a mortgage, that mortgage is then filed on record at the county courthouse. Typically these were identified by a book and page number. So if you lived in Smith County, your mortgage might be recorded at Book 72, Page 690 at the County Courthouse. The bank would also have copies of it, and often, but not always, the note would be referenced or attached to the mortgage.

So imagine you have a farm inthe 1930's and you're paying the First National Bank of Missouri $25 dollars a month. Imagine Pretty Boy Floyd comes to rob the bank and tosses a firebomb into the filing room, burning up a bunch of documents.

Does your obligation to pay the bank stop? Probably not. When you miss payments, they'll send you letters and eventually sue you.

The question is, how does the bank know? Well, hopefully the bank had backup copies. But even if they did not, in the aftermath of this incident, the first thing the bank is going to do is send someone down to the county courthouse to obtain copies (or type up certified copies, make notes, or whatever was the appropriate method at the time) of all the mortgages the County clerk has on file for the first national bank. The staff member is going to come back with a notebook of all the names of people who have mortgages on file. If they had a backup copy of the note, then they'd use that. If not, partial records like receipt books, payment records, coupons, things like that.

If you don't pay, the bank may sue you, and the question is going to be "can they prove you owed the money?" The statute of frauds requires that you have the original document in writing in certain cases. However, there are exceptions. One of those is if a contract has already been performed or partially performed. The bank will put a witness on the stand to testify, yes, on the 1st of January 1929 you executed a note, and unfortunately the original was destroyed, but there's a copy filed with the courthouse which shows it exists. Then here's the ledger notes for the 22 payments you made, showing you knew about the note.

Contract cases are "in equity" meaning you don't get a jury. A judge is the one deciding your case, and the judge knows how important the local bank is and as long as the bank followed the rules to the best of their ability, the judge probably will not accept an argument of "well, since Pretty boy burnt the original, can you really know?")

Maybe someone could fall through the cracks, but the answer is that it's pretty unlikely that someone robbing the bank could actually meaningfully impair the bank's ability to collect its debts.