Gosh, I don't know, how about literally all of the problems that 2FA solves in the first place? Passwords alone are a bad solution (often forgotten, easily re-used insecurely) for people without all of the challenges and frequent mental issues that accompany homelessness, why would you think they'd be a good solution for people who, as the OP says, aren't capable of keeping track of a physical device for more than N weeks?
I'm not unsympathetic to the problems of the homeless ant the burdens 2FA entails, but I'm also not willing to ignore the huge problems the 2FA solves, and realizing there will often be a tradeoff between making it very difficult to hack into accounts and making it easy for people with mental and other problems access their accounts.