"What is the motive then for a for-profit company to even exist? If every insurance company has to take everybody, that implies they have to also provide the same coverage to everyone at the same rate. "

Correct. And whoever is more efficient wins. That's no different from any other retail business. My grocery store sells lettuce to anybody. Some are more efficient and can offer better prices. Or they sell better lettuce. But regulation sets minimum requirements so nobody can sell fake or toxic lettuce.

I thought that's how the market works.