Yes you're right, the control lies with the owner. The owner is quite often also the CEO with a private company, but the distinction is worth clarifying.

Strictly speaking, the same is true of publicly traded companies. However with publicly traded companies there is also law that obligates the CEO to act on behalf of the shareholders. The shareholders are the owner, just like with private companies. However a private CEO would just be in breach of their employment contract, a public CEO would be in breach of the law.

Ultimately the reality of publicly traded companies means that "the CEO works for the owner" in all practical purposes is "the CEO pursues profit above all else". While it would technically be possible for all the shareholders to vote that the company do something else, in reality that almost never happens - there are too many ways for shareholders to buy into the company and say "no I want money". Thus, privately owned businesses have the opportunity, under direction from the owner/CEO of behaving differently to publicly traded companies. That doesn't mean they will, because many private business owners want to make money just the same as public shareholders, but the possibility is much higher.

Your entire premise is simply false.

No it isn't, and the things I'm explaining to you are widely understood.