At some point, though, the solution doesn't become "make it possible for anyone to access any account without any proof of identity", it becomes "make it possible to live in society -- receive medical care, eat, be sheltered -- without any proof of identity".

Proof of identity should be a government function, and that we likely have millions of people in the US with no way to prove their identity has real-world consequences beyond the flaw in my post-office-account-recovery-scheme; it affects access to benefits, as you said, as well as voting and being able to even prove your citizenship. That should be fixed too, but I'm not sure we can do any better for internet identity verification than the post office fallback.