the homeless are more akin to pests as far as the money is concerned.

I'd go one step further. Homelessness, and poverty in general, are necessary to capitalism. If the consequences of poverty weren't so bad, workers wouldn't fear losing their jobs so much. Homelessness helps maintain the authority of the boss over the worker and the corresponding hierarchy of capital over labor.

and I’ll never fully understand why they believe that will ever work.

Because usually they're safely tucked away all nice and warm in their houses or apartments, not actually caring beyond the vague notion of "oh nooooo bad things are happening to them oh nooooo..... Anyway as I was saying..."

It's like telling black person "you'll have equality someday, no need to get pushy or inconvenience anyone" in the US in 1865.

Its easy to say something like that when you already enjoy the "comforts" that they don't have.

You could theoretically have a “free market” in a system where the whole point is to take care of people

You could theoretically have a "free market" for charitable care in a system where wealth was equitably distributed and the reward for labor came from the communal surplus rather than negotiated out of the tight grip of a monopolist. That's the cornerstone of programs like Medicare/caid, HHS, and the Pentagon. A public trust pays for privately management of labor toward certain political ends.

But you need a public trust managed by a popular government to achieve this system, and we've been rolling back both under our current increasingly privatized electoral system.

I know real people who think like the people in the image and I’ll never fully understand why they believe that will ever work.

The view is predicated on this idea of an economic ladder that everyone gets to climb. And failure is attributed to sloth or incompetence rather than an absence of individualist opportunities.

Prior to the flood of cheap labor created by the Baby Boom, followed by the post-80s globalization of industry, it could work thanks to the leverage afforded to labor unions and regionally fixed capital stocks. Philadelphia steel workers have a strong negotiating hand when their foundries were the primer source of domestic steel production. If you could work the mills, you could live a comfortable middle class existence as a result.

Now, steel imports are so cheap that there's little incentive to do steel manufacturing in the US at all. Workers have no hand to play, so they're forced out of the industry. The mythology of the Philadelphia Steel Worker lives on, though.

You can try to have a totally unregulated market with the alternative goal that you mentioned above but actually that doesn't matter because if you have a small percent of greedy bastards, they're going to try to co-op the system and either you regulate them or you lose.