Have you tried playing this out to the end? You're hung up on words like "require" that I, some random dude on the interweb used, rather than actually realizing the intent or spirit of the comment. You think any company receiving negative press is going to just allow that to continue without addressing the concerns somehow? An activist investor/shareholder doesn't just stop with press briefings. If they still feel agrieved, they can attempt to manipulate the board by having their own member installed. From there, they can direct if the CEO even gets to keep their job. As a board member, they have a lot more power.
I feel like you're limiting your imagination on how much power stockholders can have. Some companies are probably a little more resielient against this type of "attack", but rich people play all sorts of games because they feel slighted or bored. Watch the TV show "Billions" if you want some fun fictional aspects, but these stories are not made in a vacuum.