Most managers I worked for in corporate setting didn’t really care for people. They were loyal to their superiors. And their subordinates were more or less tools for a goal. Most absurd situation was in my previous job where the company hired experienced manager as a developer and suppressed him on every occasion. My precursor and I left that place very quickly. So the people skills from my single data point are not the prerequisite for becoming manager. But again. I am single data point. Maybe there are places where managers care about something else than budgets, timelines and their own bonuses.
That's kind of beside the point - for the sake of argument, let's say managers only care about "budgets, timelines". It would still be a stretch to switch from an IC role to suddenly managing the "budgets, timelines" not only of a team but of a whole department
I am convinced, that the guys, who got promoted, know how to manage things. Even in toxic place I worked as an important IC I assisted doing project plans and technology roadmaps. The department things were same charts, but without technical slang and absolutely anonymized threading workers as an abstract number “headcount”.
"Generally, management of many is the same as management of few. It is a matter of organization. And to control many is the same as to control few. This is a matter of formations and signals."