It's not a caricature. These people are easy to find in many cities. They're not uncommon sights, even if they don't resemble the average homeless person.
The question was about the "severely mentally ill".
If the counterpoint is that it's rude to acknowledge the people who seem to be utterly insane, well I think that's kind of shit. If it's that these aren't the people we're referring to when we say "severely mentally ill", then I think that's just disingenuous.
A person can be severely mentally ill in many ways, not all of them are highly public and immediately offensive to passers by. So while the people you describe would indeed fall into that category, many readers would likely place other people with debilitating mental illnesses into that category as well.
That's true, but someone who is severely mentally ill in an invisible way is not going to be categorized as such by a passerby, so it's an obviously semantic ambiguity. Saying what is technically true but not understood is just distracting.
That's not necessarily true either. Someone who is catatonic might be noticed by a passerby but doesn't scream "this person isn't safe to have in a shelter".
> The definition of "mentally ill homeless" is surely nuanced and varied,
but in the rest of the post you appear to be discussing this strawman as if it is the primary concern.