my experience is that most people running large organizations got there (and are making the 250k-350k salaries you mention or even more) not because no one else wanted or could do the job, but because they were good at working the politics and self promoting. Often the best in our companies do not receive the recognition or the pay they deserve while the manipulative do. Just my experience though.
And it is even more so in the public sector.. because if there is a screw-up.. we just raise the taxes. Consequences on bureaucratic mistakes are less, so the system doesn’t ‘self-correct’ as well. Bad CEOs are removed when their company goes bankrupt of the stockholders/directors vote them out. In the public sector, it tends to be a different story — promoted diagonally.