Having a set of emergency backup codes for each MFA site that one visits is an idea that does not scale gracefully. The way things are going, with more and more sites offering or even demanding MFA, the shoebox in the attic will soon start overflowing.
I suggest that primary insurance against loss or damage to smartphone should be to have the authenticator app running on three separate mobile devices, with each instance syncing automatically with the other two whenever a change is made.
Is this achievable with currently available tools?