Management and leadership involve plenty of problem solving! It's genuinely hard work, not just faffing around - that's just the stereotype that prevails among people who have never been in a management role in the first place. For one thing, if you're in management, there's probably a higher-level manager that you're accountable to! Funny how people seem to forget this.
So the problems you solve in management are how to get people to solve problems, not the physical/logical problems themselves. It's not faffing around, and it's a higher level of abstraction, arguably something gifted people would be good at if they hadn't been trained to solve problems like a super IC.
You specifically said that management doesn't involve "talent", "problem solving", "intelligence" or "creativity". And often the best managers are people who are just as familiar with the physical/logical domain as the concrete "problem solvers" - after all, much of the difficulty of "getting people to solve problems" is things like knowing when to call BS on someone vs. when there's an actual issue that's gumming up the work and that one should deploy resources on. Stuff that's way too hard unless you actually grok what's going on.
