I think it would be fine for the library to have/be the 2ND FACTOR and the user would still need their password. Being at a physical location seems like a reasonable 2FA (more reasonable than a phone in these cases).

Could the library buy a few FIDO tokens, hot glue them into the backs of the computers, users add them as 2fa to their accounts and now the computer being wiped between users is no longer an obstacle?