Clearly, if you a CEO and publicly supporting causes, your support for those causes will have an impact on the reputation of the organisation.
Now, I sure you can think of examples of more extreme spare-time beliefs that you might think were incompatible with being CEO, so its a question of where you particularly draw the line. (Or perhaps you can't and you think that any extreme free-time campaigning is fine).
FWIW, I'm not in San Francisco.
Now Mozilla has, whether you like it or not decided to build its brand values and ethos around equality and inclusivity - it is written all through its positioning and marketing material. Similarly Apple pitched privacy. If it was discovered that Tim Cook had a part time gig with Cambridge Analytica etc. I doubt he would last that long.
Regarding Google et al:
> Just because they don't explicitly say it out loud doesn't mean they care.
That's correct they don't say it out loud, they don't explicity campaign on the issue and their CEO doesn't donate money towards it.
> I trust Brave and Brendan far more on privacy issue since they only care about that and don't shove woke politics onto my face.
If Brendan started donating to an organisation campaigning to give law enforcement and marketing companies access to your personal data, do you think it would be compatible with his position at Brave? Bear in mind that, apparently when it comes to privacy, very few people care about that shit (as can be seen from Chrome's market share) and there is good money to be made selling data.