Hmm... I'm not convinced. I think you want john carmack as an engineer with a lot of freedom and resource and a limited amount of management responsibilities. He seems far better suited to being first among equals (first among equal engineers) than to being a technical business-decision maker. Give him the role of arguing for the right decisions rather than making them, and I think you'll get a lot better value for your money.
If there's a man you want as CTO, it's Carmack. One of the all time great programmers working full-time on VR is the best possible outcome.