When someone goes bankrupt most don't say "Oh this bankrupt person tried to [blah blah]."
It just seems weird to me with homeless we replace their entire personality with the trait.
The following sentences say the same thing but I feel they come across vastly different.
"The mentally handicapped person went to the store."
vs
"The person, who is mentally handicapped, went to the store."
With your examples about mental handicap, the same principle applies - it matters what's the topic of the discussion, what's the next sentence, and thus where the emphasis should be. If the fact that that person is mentally handicapped is irrelevant to the rest of the story, then "person" is the part that should be emphasised; but if the mental handicap is the key factor for what happens next, then that is the most salient information of the sentence and it should be emphasised.
Describe the man, don't define him.
First one is obvious, every person is ... a person. 2nd and 3rd are not determinable by looking.
Anyways, I can see this is pointless. I hope your day is as pleasant as you are.
Since we're firing off arbitrary primary characteristics, let's just choose "male" for me. Could be that I'm homeless. Could be I'm a billionaire. I could be a Nazi with heads in my freezer.
Dimunitive language dehumanizes. Why the effort to classify with a single word?
Reduce redundancy. Other "occupations" do this as well. You say "painter", not "people/person who paint(s)".
Two, if you really want to reduce redundancy just use "person/people." Everyone knows if you say "homeless person", that that person is, in fact, a person. Seems like an extra word there.
If you want to be precise and point out someone is homeless, use "person, who is homeless." If two words and a comma is too redundant for you, see my first point.