The new CEO accepts the job.
The chairwoman meets with him before the first board meeting to impart some wisdom. "One of your most important jobs is to make sure you develop your eventual successor."
It sounds so sagely. The CEO says, "Yes. Absolutely!" But, in truth, it's the furthest thing from his mind.
One of the worst things a Board of Directors can do is abdicate their responsibility and put the current CEO in charge of hiring the next CEO. Yes, a good CEO will fully develop the leaders around him. Then, there will be someone who can step in take the helm when it's time for the exit.
Realistically, most CEOs are constantly concerned they will be shown the exit long before they are ready to do it on their own. This creates an incentive to make sure there is no one seen as their heir apparent.
Even when the CEO has done an excellent job and is ready to retire there is a natural tendency to make sure the next CEO is not viewed as better or who will undo all his work. The company gets a lesser version of the current guy who is unlikely to make needed changes.
CEO's need to focus on their job of running the company. Boards need to do theirs which is ensuring the company's long term viability.