Hell, it should be trivial to offer federal government provided emails with ID verification with customer service in the event of loss of device/loss of ID/death/etc.
Many other countries have a central government portal with secure messaging, with federated identify. Heavily reliant on 2FA of course.
0The issue here is that homeless don't hold onto anything physical for 4 months; identity verification breaks down in-person immediately as shelters/libraries can't be expected to run a facial recognition operation, and specific shelter employees/volunteers aren't guaranteed to be there anytime a homeless person might walk in and need those backup codes, but it breaks down even further online since 2fa is inherently 'what you know' + ('what you have'/'who you are').
The point is the hardest part of the problem is already solved - which is the physical infrastructure and labor. As for not holding onto physical items, USPS also has little boxes that people can keep their belongings in.
To be fair I don't see how any government system can do better regarding identity on the internet. Login.gov is one of the best services I've used for access to usajobs/SSA/etc but it follows some of the same security best practices people are complaining about here with no real way to re-gain access to your login.gov account should you lose your 2fa methods (afaik).