Yes, unironically, they should. And most would be trying to do exactly that.
I don’t know why some people treat economic status as some immutable property outside of your control. People move up and down in economic status all the time. And most people move up as they get older and get more work experience and higher paying jobs.
Having a stable place to live with a physical address instead of a tent, and possibly being around other people who are trying to improve their lives instead of a bunch of drug addicts would absolutely help people “not be poor”.