Granted, hospitals and health systems most often aren't about maximizing shareholder value. So I'll grant that point, although I argue it's a semantic one, to a large degree, because what are they about? Fulfilling their mission to provide care for those who don't have the means. Providing great patient outcomes, hopefully. Providing great patient experience too, so those patients will tell others about their great experience. Isn't that a form of maximizing shareholder value?
CEOs at nonprofit hospitals and health systems still have to run their institutions like a business. The old saw "no margin, no mission," didn't come from nowhere, after all. In fact, at nonprofit hospitals and health systems, it might even be more important to maximize the margin, because you are pretty much required to lose so much money every year on a certain group of patients, or a certain group of services.
So what does that have to do with whether you should stay on in some sort of leadership role after you retire?
I think it depends on the person. If the former CEO is gone, there's no worry about whether he's overstepping his bounds by having weekly chats with his successor. There's no worry on the outside of the organization that he or she is really pulling the strings behind the scenes. There's no concern that the physicians or others in the organization won't see the new person as the "real" leader.
It seems there are fewer opportunities for problems like this if the old guy leaves everything up to the new one.
Ride off into the sunset. It'll probably better for everyone involved.