He's not impervious to criticism, he just thinks that he is. Massive negative press following massive negative press about his handling of the company and the loss of revenue is not going to make his shareholders happy, which is really what all he'd care about.
For Twitter at least, he doesn't have shareholders. He has investors, many of whom would like nothing more than for Twitter to die so that their citizens can't use it to coordinate political action against their regime.
He can pay to surround himself with yes-men all he wants, but that just makes any criticism that reaches him sting more. You can see it in action by the petty moves he makes against people and things he doesn't like.