Chief surgeons do all those things you describe. Check out the job description [0] as they are basically department heads doing recruiting, budgeting, supervising, etc. Chief Surgeon doesn’t mean best surgeon.
But it is a bad analogy, as you've implied. What I was trying to get across is managers should be specialised, they should be someone who might or might not have been a programmer, but is now specifically managing people, or projects.
Managers who code, do exist, as to engineers who manage. But its better at large companies to have specialists to manage projects and people. With support, training and performance management, you tend to get a better outcome.
What’s interesting is that all chief surgeons were once surgeons, probably really good. So there’s idea that you have to do the thing to manage the thing.
Medicine is strange in many ways, but I think this isn’t a bad practice.
I’ve worked for non-dev managers and sometimes they are good. But I prefer to work for someone who at least once was a dev as I think they understand better what is possible and how to coordinate and lead.
When I managed teams I always tried to do something useful in the codebase. I don’t think it is possible to manage well and work deeply enough to make great software. But it’s possible to at least be able to do my own builds and at least test out different techniques.
[0] https://www.betterteam.com/chief-surgeon-job-description