Public listings or lack thereof are orthogonal to corporate legitimacy. Cf. Enron and SAS.

If you're a profitable private company, aren't you legit?

In some sense, yes, but check out Sarbanes-Oxley: https://en.wikipedia.org/wiki/Sarbanes%E2%80%93Oxley_Act. Public companies are under much more scrutiny in terms of compliance, and they have a certain type of pressure to post good quarterly results that private companies don't.

What is the incentive to go public in today's environment other than an initial fundraising, and an unpredictable monetary bonus for employees?

Edit: Appended another question.

That is how most investors cash out and turn a theoretical return into an actual one.

It also makes the value of your corporate equity much clearer which can enable acquisitions.

Depends. I think with the increased scrutiny as a public company, you have much less wiggle room for creative accounting tricks.