I have learned that the purpose of a CEO in many (most?) companies is not to actually run the company.

Their purpose is to be the human delegate that represents the corporate entity. They are empowered to act on behalf of the entire organization. This is important because a corporation of many individuals generally cannot act as one. You can sometimes get close with really, really good teamwork, but that's still not enough for a lot of customers.

If you are in some regulated B2B context, not having a CEO or having a "non-traditional" CEO can result in lost deals. The counterparties are looking for "one throat to choke" when they are being made responsible for approving large contracts. They want a person's cell phone number they can call when they're having a bad mood about whatever they've just paid for. They don't want to talk to your developers or the cranky old founders. They want to talk to a person who can be summoned to any arbitrary golf course within 48 hours and who will arrive in a fantastic mood 100% of the time. This person often doesn't have to know a goddamn thing about what your company's product actually does. They just have to be a delegate. A conduit of information and a representative element.