I agree. Governments just don’t care (in the actionable sense) about homelessness except when it gives rise to disruptive or ‘criminal’ behavior, so they tend to associate it with those things.
TBH I think the gist of the “mental health” claim stems from the massive amount of Vietnam vets the Fed unceremoniously abandoned to the streets. To claim these people were homeless due to mental health wouldn’t be totally inaccurate, but it definitely buries the lead. More than 1 in 10 homeless people are vets in the US.