"Capitalism isn't defined by markets, free or otherwise. Capitalism is defined by the private ownership "
Agreed.

"I don't need to own a home, as long as I have a place to stay where I'm not bothered by others. "
How is that not ownership?

"Again, capitalism is defined by the private ownership of the means of production, socialism is defined by the workers owning the means of production; "
Those workers are still private individuals, it is still private ownership. Just becouse it is multiple people does not make it not capitalistic.
As for free markets, theres some that are pretty close in the darkweb. Very consumer friendly, when my goods didnt arrive in a reasonable amount of time they just sent it again. I ended up getting both on the same day somehow.

In any event, you seem to draw a weird distinction between workers and owners, between private and personal property which really doesnt make a lot of sense. Capitalism is the workers right to the result of their labour and/or fair compensation for their labour/goods.
The elites are lying to you about capitalism.

Lets say, theoreticly, a lawyer who has a sister who is a doctor was to talk about how universal healthcare would require his sister to be a slave (or other people to be a slave to pay for her if she is willing to accept that and if she isnt then... well then she would have to be the slave ,wouldnt she?) but completly ignores that there is no free market for healthcare. Im not allowed to make cheap insulin and sell it for little more than production costs precisly BECOUSE it is allready socialised. Likewise, im not allowed to treat patients unless ive got a license which in the US would require me to go through their expensive education programs.
Whereas in China theres a lot more of a free market here, with so called barefoot doctors that arent educated for the most basic of needs, you dont need to see someone who went to medschool for years to deal with the common cold.

Again, you dont take issue with capitalism. You take issue with socialism. Or more precisly, with keynesian economics.