You and your parent comment are both right. They certainly are different skills, and yes, some people possess both sets of skills.

You're just restating the same thing again.

Why is a business coach different than other kinds of teachers?

Going further, why should you learn anything from any kind of professional teacher? By your logic, doesn't the fact that they're teaching it imply that they can't do it?

A professional teacher is different, particularly at the primary/elementary level. Of the teachers I know and have worked with (I also have a few in my family), they do it because they enjoy teaching and would not choose to do anything else.

Business coaching is something you do after running a business.

Teaching at a secondary you will often find people who previously worked in industry. My high school chemistry teacher for example previously worked as an industrial chemist which put his level of expertise way above what was required.

And at the tertiary you will get a mix of academics and tutors, who are either actively researching or working in their field.

Which is why mentoring groups with other business owners can be a lot more beneficial than a coach.

Some coaches aged out of it, but many didn't. And yes, I do know that the idea is so widely held that there's a silly simplistic quote for it.