The administrational/bureaucracy seems to generally be a problem with the US healthcare system, even for more affluent people. But for homeless people it needs to be close to zero. For moral reasons (it would and does effectively prevent them from being treated like the humans they are) but also for efficiency reasons. If there's clearly no money to be had from someone, minimize your handling losses by giving them what they legally deserve without making it a lot of work for everyone.

> minimize your handling losses by giving them what they legally deserve without making it a lot of work for everyone.

The problem necessitating bureaucracy in a separate system for the poor on top of the US’s mostly-private-pay health care system is identifying who is eligible and what they legally deserve.

Other countries define "what they legally deserve" that pretty much as "what makes sense, medically"...

The part that requires bureaucracy for Medicaid is making sure the recipient is legally entitled to Medicaid at all (and, if they are, that they still don't have other insurance which should be paying ahead of Medicaid.)

That’s where having a separate system for the medically indigent in a mostly prvate-pay system becomes much more burdensome, per covered person, than a universal coverage system without private insurance.