All a question of tone and style?
When someone leaves the company, they naturally take a significant portion of their own personal brand value with them. Naturally, a large amount of resources from PR and communications departments flow into the personal brand strategy of a member of the company's management. In fact, they have to, because this presence is, as above, part of their public image.
Personalities from the board and C-level are never just present as individuals in social networks, but always as official representatives of the company. Therefore, every small statement can have a direct impact on the company, even increasing or decreasing its (stock market) value.
The extent to which the personal profiles of decision-makers bank data on social networks have economic and economic policy consequences can be seen everywhere. People like Elon Musk (Tesla) are extreme examples. Even less eccentric and famous CEOs must and should think carefully about what they say. As an exposed decision-maker, you are never out and about in private, and certainly not on social media.
Therefore, the tone and style with which all parties communicate in a particularly sensitive situation such as the removal of a CEO will determine how the situation affects the company as well as the reputation of the people involved.
Why Social Media Is Not Optional
But anyone who believes that visibility on social networks increases the value of a CEO's personal brand is putting the cart before the horse. The leading minds of large companies have been in the public eye, and not just since the invention of LinkedIn.
It should actually be self-evident that this creates an obligation to use media in a contemporary manner. In reality, in my opinion, such a presence is not optional and certainly not a special additional service on the part of those concerned. Rather, it is a requirement of the times.