But "untreated mental illness" isn't the same as "mental illness that requires institutionalization" which is what the OP is saying.
Additionally, a lot of mental illnesses can be reasonably managed with proper medication, and in my mind very, very few actually require institutionalization. But we as a country can't even get behind the idea of universal healthcare for non-homeless let alone homeless people. Somehow institutionalizing them seems more feasible or reasonable than just covering their medical care?.. I don't get it.