The best solution I can think of that doesn't compromise security is hardware keys. GMail has very good FIDO support. The keys are easier to use than TOTP and vastly more secure than SMS. They do not depend on any phone or phone service, and there is no transferring necessary at any time. The librarian would just need to get the person logged in successfully one time, get the key attached to the person's account, and…
Reading this comment, I thought Yubikeys, which aren't /expensive/, but aren't cheap either. I was pleased to see they have a key targeted at this specific use case now - the Security Key Series [0]. At $25, that is not too bad a price, and something I'd buy for the members of my family without much hesitation. The hangup with this, which I think the librarian in question will feel, is what happens when someone loses…
This compromises security somewhat, since the library houses one of the second factors, but IMO it's preferable to total account lockout (and still superior to SMS verification).