Private or public ownership

I'm constantly realising basic things I don’t know about economics.

Here’s one. Why do we have private firms? Shouldn’t public organisations be better because they don’t have to make a profit so all their takings can go back into the enterprise? and because they can be run entirely for the public good and not for private profit? Just think of those cases when public/private are clear alternatives, such as utilities and railways.

Is it because public organisations are inherently inefficient (badly run) that they’re so out of favour? And that it necessarily takes the profit motive to get people to run things well?

Public companies (which aren’t of course public in the sense that the NHS is -- or was) have shareholders. The money people have paid for the shares is what the company runs on, as well as its takings. The reason people buy the shares is that their value may grow and that the company pays dividends. A public enterprise doesn’t have to pay dividends.

A public enterprise does, however, have to have money -- the equivalent of the money a company gets from the sale of its shares. The money comes, most likely, from loans on which the interest is paid out of the takings, if any (from train fares, for instance) and taxes. (A company, too, of course, may borrow money, on which interest has to be paid, but for simplicity let’s leave that out of account.)

Are companies deemed to be more efficient because the sale of shares does away with the need to borrow money and hence the cost of interest? and because the cost of paying dividends is less than that of paying interest?

If I don’t understand stuff like this, the case for economics in schools seems clear.

If someone enlightens me, I'll gladly post their corrections.