In an actual free market the distinction between “public” and “private” company would be really just some transparency/accounting standard or set of standards set by the particular exchange(s) that the shares were traded on.

And just to be clear in practice, stocks are not priced based on their voting rights but rather expectations of their future cash flows (dividends) or simply how much the stock is expected to go up or down by. A lot of modern financial activity is nonsense.

If you really want to actually understand what a company is you should go look up when corporations became a thing and why. It had to do with various legal protections to stimulate economic activity and tax revenues.

All of these modern notions and principles that you’re invoking or basing your arguments on about ownership are most likely just contrived nonsense. These aren’t necessarily eternal principles of anything.