This is a question that really struck a chord with me. I went down the business route in my 20s, and then, much later, went to a catering college and trained as a "chef" as a career change (hobby?).
Whilst it's necessary to know how to run a business if you want to own a restaurant or even manage one, the skills are very different to being able to cook in a professional kitchen. Being a business grad and a good cook won't be enough. Experience in a professional kitchen is essential if you want to even function as a commis.
So, I would ask "do you want to become a chef or a restaurateur?". If you want to become a chef, then you'd better focus on the culinary side - it's not easy, whether college-taught or self-taught. If you want to be a restaurateur, you'll have to know about business.
Anyway, some of it is a bit daft. You can go to college for a year or two, get a piece of paper to say you're a chef. But you're not. You're the lowliest of cooks. It might be better if there was a qualification that gave the entitlement to call oneself a chef.

(Looking forward to howls of disapproval of what I just wrote.)