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.