Where would a company ever get their market cap in cash? If they had that, wouldn’t they by definition have a higher market cap, since the value of the company is cash + the rest of the company?

> since the value of the company is cash + the rest of the company?

Failing companies sometimes trade below cash value. OP's basically creating a rule by which only failing companies are allowed to go public. (Or those who have paid a king's ransom to a megabank.)