The statement "use less of their brains" does not necessarily do managers justice. Managers are good at (guess what) managing people, and that takes relationship skills that some people might not have or not good at. Having being both a software engineer and a manager, I would say it takes some special skill to do either work, who gets paid more is up for debate but not necessarily that an engineer should be paid more because they use their "brains" more.
Another example is a CEO who is non-technical. What skills do they have that make them have higher salaries or even bonuses? Guess what, they can make deals happen. They can secure a funding round, they can close an enterprise deal and it takes some special skills to do that as well.