No, it isn’t a problem if the a CEO of Twitter doesn’t care about their tweets. The proper task of any CEO is maintaining the company as an immortal entity.
If the general public cares about the CEO’s tweets, then necessarily there will be a danger that the death of the CEO will ignite a crisis in the general public; e.g. Steve Jobs at Apple.
I never suggested the CEO needs to be the star tweeter. What I am saying is that the CEO needs to be a tweeter, i.e. personally invested in his/her company's product. Companies with CEOs who don't give a shit about about their product/users tend not last long as "immortal entities".
If the general public cares about the CEO’s tweets, then necessarily there will be a danger that the death of the CEO will ignite a crisis in the general public; e.g. Steve Jobs at Apple.
The producer is not the star.