These are all problems. On the other hand we know for sure from centuries (millenia?) of evidence that allowing the people who most crave power to be the ones who seek and obtain said power, is a recipe for disaster as well.

It is interesting to brainstorm what could be a better system.

I think clear qualifications should be required. So to use the HHS example, it would have criteria to be a doctor, certain years of practice, perhaps practice in several specialties for breadth, some research experience, some administrative experience. You'd want to make the criteria narrow enough that only mostly truly qualified people meet it, but also broad enough that at least tens of thousands of people do qualify for the role, so that someone can't just easily fine-tune their resume to assure themselves a spot. It needs to be impossible to game.

I don't have the answers, but clearly we can do better than the current system where only power-hungry psychopaths choose to travel the road to obtain the vast power they crave.