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.