Let's just presume that all of us do something from time to time that the mob might find unpalatable: we cheat on our spouse, we evade taxes, we contribute to unpopular causes, and so on.
The question here is basically "What the hell are we doing?" Do we want a world where a few folks can research random corporate CEOs, dig up something to stir up the mob, then end up getting him fired? Because you're absolutely right about the role of CEO: once he's smeared with so much mud, he becomes a terrible liability for the company. The question is whether or not we're going to allow current political passions to influence where we start slinging the mud. You can't on one hand participate in this kind of attack on this guy because he was "wrong" and then on the other hand proclaim that sure, he has freedom of speech, but we also have the right to punish him. Or rather, you can, but it just basically means you can punish whomever you like.
And the end-game here is terribly clear as well: young leaders in tech will learn that it's completely fine to hold unpopular opinions; just don't ever let them out to where they can be observed publicly. This destroys open public discourse and creates a society where there are certain things you can say and certain things you can't. If you cross that line we will hunt you down and econmically punish you. That's crazy.
An alternate moral frame for the same concept: do we want a world where social pressure can be used to make particular political actions risky for people who are otherwise immune to or insulated from risk? There are precious few other options for discouraging wealthy, influential people from disproportionately wielding their wealth and influence in the service of reactionary ends.
When it comes down to it, I think Eich's supporters and detractors mostly differ on where the line should be drawn, not on whether there should be a line at all. There is unquestionably some level of private expressive action by a CEO that would be almost universally acknowledged as incompatible with holding the position.
If it were to turn out that Tim Cook was a die-hard white supremacist who had donated repeatedly to the cause and appeared on video speaking passionately about the need to ethnically cleanse the United States, I think you'd have only a tiny fraction of holdouts defending him on the basis of free speech. Similarly, if the CEO of BP was ousted for being critical of the treatment of women in Saudi Arabia, I think you'd hear rights-based arguments against it from people who reject such arguments for Eich.
I was one of the signers of the CREDO petition calling for Eich to step down. I wound up regretting that -- and I had some second thoughts even at the time -- but at the moment I signed it I had not yet read his statement pledging to uphold Mozilla's diversity policy, along with his claim (which I have never seen contradicted) that he treated everyone at Mozilla with respect. Had he failed to do those things, this would indeed have been an easy call.
Should say, a CEO be free to donate to a white power group and not have it affect their career? What if they donated to a group trying to have black people declared as property? Would it be 'destroying open public discourse' or 'crazy' if some of the future property then protested or boycotted said company? If I'm understanding all the arguments of this type correctly, the answer appears to be "yes".
Mozilla lives and dies on its ability to attract and retain the best talent. Eich as CEO was only useful if he could support that end.