Rather than firing a CEO, he may do a good job as CTO or CFO or even a General Manager(GM). Why is it always 'get out' for CEOs?
It's not just CEOs. It's pretty unusual for anyone in a leadership position to step down and take a lower role in that organization.
To start with it usually makes for a strained relationship between the former leader and their new peers. If someone didn't get on well with the former leader they may try to take some sort of revenge now they are equals. Or simply decline to cooperate. Or other bad things.
Secondly if makes things awkward for the new leader. It can make for problems to have someone around who knows the organization better than you, who people are used to taking orders from and to whom the team probably owes a lot of loyalty. It's also going to make it harder for the new leader to make significant changes to how things are done. CEOs in particular are often fired because the company wants a major change of direction. In the worst cases where there is a lot of politics (very frequently true at the C-level) a newly fired leader may try to actively sabotage the new leader in the hope of getting their job back.
Also, CEOs rarely have difficulty in finding new jobs, and usually at better pay than they would get in a lower position, so there isn't much incentive to stay.
At any career level, having someone replace you, and then having to witness daily how that new person performs at your previous the job (either better or worse than you), would be extremely frustrating to most folks.
Further, the new boss doesn't want to deal with the baggage and office politics of a disgruntled employee who has been demoted.
Finally, as you might imagine, CEOs are incredibly career motivated. They probably wouldn't even consider stepping down - it would be step back in their career.
Better for everyone to just part ways.
In most cultures/organisations, demoting is extremely rare, except perhaps as a step towards dismissal. Demoting someone is extremely demotivating, the assumption is that this person will be resentful, have trouble accepting instructions from people who were once peers or subordinate, will have no reason to assume they have a shot a promotion, etc.
Whether for a CEO, a middle manager or just an individual contributor, it's much more common to be dismissed, to be put under pressure or to be deprived of meaningful assignments to provoke a resignation or to be “promoted” to a well-paid but less strategic position where your career will stall than to be demoted.
The competition at that level is incredibly fierce and, in order to win one of these coveted positions, it's generally not enough to be good at your job. You have to play a lot of dirty politics which often involves stabbing people in the back. If a CEO were to get demoted and he opted not to leave the company, there's simply no way his successor would be able to trust him to be loyal and to respect the new hierarchy. He would do everything he could to consolidate power, and force his way back up to the top. The company could end up split into warring factions which would be disastrous for all involved.
I would risk to say the job of some CEOs is being fired in the end.
I have been at least in two organisations bought by a larger one, that when they wanted to pass unpopular measures/fire multiple people they hired a CEO for them to be the face of those measures.
And in one of them, the actual interim CEO, a partner, stepped down and went to his former role.