I've heard from several people who knew James Watson personally that he was completely off his rocker. He lost his position because he was a total embarrassment, this wasn't the PC police at work. No opinion about the other names.
Summers lost his job as university president because he alienated a significant portion of the faculty by making dramatic political decisions without buy-in from the rest of the community. (Firing popular administrative staff, diverting budget from long-running projects with internal political support, getting in aggressive fights with influential faculty members, etc.) Overall he made for a divisive and relatively ineffective leader of the university. His poorly considered sex differences comments were just a convenient excuse to put someone more politically savvy and charismatic in as a leader.
Additionally, it’s hard to feel bad for him as the way he “lost his job” was by being offered an extraordinarily prestigious named professorship (see https://en.wikipedia.org/wiki/Harvard_University_Professor), giving him the ability to work on pretty much whatever he wants for the rest of his life, with a great salary, benefits, etc., and with few required duties.
Do you think he would not have been fired for his comments if he had been popular / well-regarded in his role as president? What does it mean that you can basically be terrible at your job but not get fired until you express a minor deviation from orthodox thought?
Yes, that is correct. I don’t think the same comments would have gotten him removed as president if he hadn’t already otherwise lost political support from many in the faculty. Obviously even a widely supported and popular university president making such tactless comments would have still created a public firestorm, but I don’t think it would have been an insurmountable problem.
I think he would have been removed (or removed himself) as university president sooner or later regardless given his other political blunders, but it might have taken a few years longer. As I mentioned, the comments about sex differences provided a convenient excuse and rallying point for his critics.
I wouldn’t say he was “terrible” at his job. Naïve, undiplomatic, and bull-headed, with good intentions but without enough political skill to persuade his opponents to follow him or enough empathy to understand their objections and moderate his positions.
Disclaimer: I was a first year undergraduate at the time, and my understanding comes from talking to various people at Harvard during and after the controversy, including both critics and supporters of Summers. I wasn’t well enough connected to be the ideal first-hand source about this topic.