Because it's more informative. "a person" doesn't tell anything descriptive at all; "friendly" or "a good father" would be relevant if we were talking about their personalities, but if we're talking about their economic needs, then "homeless" is the single property that's most relevant to the discussion.
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.
Then what is wrong with "person, who is homeless"?
Describe the man, don't define him.
>How about being a person? Or friendly? Or a good father?
First one is obvious, every person is ... a person. 2nd and 3rd are not determinable by looking.
So then let them be people. Why do you need to distill a person down to one adjective?
Anyways, I can see this is pointless. I hope your day is as pleasant as you are.
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."