You're obviously better off with skills. I never said you weren't. But you don't absolutely need them to be an effective manager of other people's efforts. In fact one of the best ways is to just get the hell out of their way.
One of the fundamental purposes of management is to help other people get the hell out of your way. In software, the purpose of a manager is very often to stare down requests and reject them on your behalf because you're busy doing something more important.
That is a skill.
A skilled profession is one that involves skills, i.e. things you can practice at to get better in. Retail clerks and fast food workers are generally not considered skilled professions because no matter how much you practice or improve, the customer will generally not care. Management or even something like professional basketball is a skilled profession, because there are large differences in the performance of workers based on their level of experience and practice.