Why do you say that Carmack is in management? I don't get the impression that he's a people manager. A CTO does not necessarily have people reporting to them in a people management sense. From what I've read, I would assume he does not. (I could be wrong - I don't really know.)
High judgment individual contributors often take on responsibilities that can be considered management, like deciding business and technical strategy, designing products, prioritizing roadmaps, etc., but while these are management functions, taking on these responsibilities does not mean that one is in management.
From the blog post, it sounds like Carmack is a highly productive, high judgment individual contributor with the responsibilities you'd expect of a CTO (technical strategy). I would say that someone is "in management" when their chief function is managing other people. From this post, Carmack seems to be delivering work as an individual contributor and (very senior) technical lead.
Along the same lines, I recommend we discourage phrasing like "move up to management". Management is a different job, not a better or superior one. In well-run technical companies there are managers and individual contributors at all seniority levels, such that one does not need to become a manager to "move up", even to CTO level.