It's not just the affair, it's why the entire incident indicates.
The role of CEO is to maximise shareholders value while running a company within a legal & regularly requirements.
This incident demonstrates:
He doesn't make wise choices
He doesn't think company rules apply to him
He lacks moral creditibility
He is also very stupid - his one the spot thinking is beyond stupid - if they had styled it out it would have received far less attention.
I don't want a reckless, slow thinking idiot in charge of my financial returns. He's not the person for it due to his character. It's that simple.
The role of HR is to protect the company (especially from situations like this).
By being involved in one of the biggest HR issues that company has seen and within just 7 months of working there also indicate that the Head of HR isn't up to her role either - believes compsny policies don't apply to her, doesn't respect or believe in professional boundaries and either doesn't understand or disregards behaviour that may have an impact on decision making, influence etc.
Neither of this people are the correct person for their respective roles. While an affair demonstrates questionable judgment, the fact that neither of them cared about being caught, zero discretion shown etc - shows their stupidity and confirms they are completely unsuitable to the role.
If the company don't want or need him, it's a great way to get ride of him. If he has a value that can't be replaced, I have no doubt he will be back working for them "indirectly" in a consultancy capacity via ab "one paper" third party.
She won't be back at all. Her position is untenable but she will fight it every step of the way and will be wanting a settlement and will get one too with a NDA.