I’m not expert on homelessness. I was just relating anecdotes which seemed responsive to the OP.
My roommate was a well-known real estate agent and professionally dressed. I don’t dispute that they may not have trusted him, but in that case I’d expect a more vague “no” rather than the unanimous response he always got that he was not offering enough money to take them away from their panhandling. Maybe that was just an easy excuse for them. I’m not in a position to know one way or another.
I’ll grant that many of these had mental problems or, at the very least, seriously disfunctional backgrounds. However, they were functional people who managed to get money, food, and shelter on a daily basis and refused offers for “better” situations. They did not regard homeless shelters, halfway houses, or training programs as improvements so they were explicitly choosing to remain in their situation. Since the only evidence of their mental problem was that they did not avail themselves of the improvements other people expected, there’s little that could be done to force change on them. It’s a hard life they chose, but they did everything they could to remain in it.
Actually, in Berkeley, it was kind of like hanging out at the mall. They may very well have had problems at homes (like any teenager) but their playing at being homeless was not an escape from that, just a slacker way to spend the day.
I’m really not trying to sound heartless and I recognize that there are needy homeless out there who deserve our help. I think that side of the issue is generally accepted and I’m just relating a couple of contrary points.

We loaded our stuff into the car, and it occurred to me that sometimes when cars run out of gas they can be hard to restart, and perhaps she did not have a gas can. I thought she might need a hand. So I started to look for her. I spotted her in the next row talking to another gentleman; still I did not connect the dots. As I approached from the rear, I could see that she had a wad, and I do mean a wad of money in her left hand held behind her back. Mostly $1s but I did see a few $5s and I think a $10. I would guesstimate that there was not less than $50 in that wad. At this point even thick as brick Rick got a clue. And I got pissed. I was conned or defrauded if you prefer, by her lies. She wasn’t out of gas, she was just begging money from people by lying. I didn’t (and still don’t) make enough money to give it to liars or con men/women.