Yes, however, we don't have good managers (due to lack of maturity or experience) to fill the job description you laid out. If you take a survey of people leaving teams or companies, you will find that many people leave due to bad managements.
I supposed that's why management is paid more - the skillset for a good manager is rarer than that of a good developer, even if they are less critical.
Well, but they pay the bad manager's "more" too, which is bad !