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.
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.