The librarian would just need to get the person logged in successfully one time, get the key attached to the person's account, and get the key onto the person's physical keychain. The librarian would also print out the backup codes, laminate them, and put them in the person's wallet. Once that's done, this particular library patron should have few authentication issues going forward. Assuming lots of repeat clientele, the auth night mare will largely end once everyone is setup.
Yeah, the key and codes might get lost, broken, or stolen, but that's the best you can do. If the person lost their actual keys and wallet, they've got bigger problems.
The question really is who is going to pay for these keys? They're a lot cheaper than phones at least, but not cheap enough.