Elon has said many times that there are no plans to take SpaceX public because the investment timeline for profits is much too long (decades). But I think the real reason is that he wants to maintain as much control as possible. I agree with him, SpaceX should not be a public company.
I agree unfortunately for us small time investors. Actually, I think Tesla going public so early in its evolution was not something Elon wanted, but was done due to pure necessity (they were about to go bankrupt and Elon didn't have any more capital to infuse).