Actually what would happen is taxpayers would have to pay more to recover the data and go through the physical copies.
So in reality we would be in more debt. Life isn't Fight club. (Movie not book)
Nope. There are plenty of backup records for all of the debts. Physical records of deeds, contracts, etc. And it's possible much of the data could have been restored through backups and data retrieval methods. What this would have caused would be delays and difficulties in handling existing mortgages but the biggest impact would have likely been an inability of people to apply for new mortgages using fanny mae, which is hardly a win for anybody.
The way the US economic/legal system works is that banks can just not pay their debts and the banks' owners don't get hurt.
If debts owed by banks aren't legally enforceable, debts owed to banks shouldn't be either. To put it another way, limited liability, if it exists, should go both ways.