...but I'd be 20x more keen to take a job at a company who doesn't fire on suspicion. E.g. a company who stands their ground and says: "We know we're subject of a PR campaign against us, but we respect the presumption of innocence: We won't fire X because he hasn't yet been proven guilty in court, and we will fire any that is proven guilty." i.e. GitHub, Mozilla both have fired CEOs on unproven claims.
A CEO's job is to champion the interests of their company. If they're mired in controversy, legitimate or otherwise, their ability to do that job is compromised. Sometimes, it's the right move for the company to stand by an unpopular CEO. Other times, it's better to walk away.
Customers and investors are fickle, and often irrational. Supporting a CEO that is driving both away because "presumption of innocence" might be the worst possible move you could make, depending on the situation. In fact, I'd argue that a hallmark characteristic of a good CEO is someone who can say, "My continued presence is hurting the company. I need to step down, so everyone else can succeed" if that's what's needed.
Yes, even if the controversy is completely made-up bullshit.
In the long run, the company and the (former) CEO will both be better off if they can look back at a gracefully handled departure, where everything turned out to be so much noise, than some Pyrrhic victory, where you've alienated customers, investors have walked, but, by god! you stood by your CEO!
Hoping that it will fix the perceived problem is exactly the mentality of everyone who attacks anything justly or unjustly for any reason thats why they're doing it.
What about proving the felony in court and make him pay for hurting shareholder value?
The upside of following courts and presumption of innocence is, you can tell your next CEO, "This won't happen to you because you'll respect the law, and what's more, if you respect the law we'll even respect the presumption of innocence and keep you in your job much longer than at Mozilla's and GitHub's. And you'll be clean of controversy when you leave, so it will be much easier for you to jump on an even bigger CEO position."