I, for one, sometimes miss the knowledge that an MBA might give me. I'm a technical CTO and I have some trouble with long term planning, budget, managing people, etc.
Managing people is hard and it’s unlikely you will learn how to manage people from an MBA. Managing people is an experience thing.
If you’re bad at managing people, my advice is that you hire people who are good a managing people and have those people report to you. At the same time, you probably want to minimize your number of directs.
You might even consider keeping your org small and have a VP of Engineering report into someone else on the C-Staff for all things execution related.
This will allow you to focus on your strengths which hopefully are technology roadmap/vision and selling that vision to customers.
When I got my MBA there was a case study(it's like a lesson chapter) I read about a top performer who didn't get along with a team. We never got to that one and it probably wasn't a favorite of the professor. But we did do a lot of cases about reallocating assets across a conglomerate or relaunching a fifty year old product or what have you. In retrospect, any insight in how to handle that troublesome high performer would have been really helpful, particularly at the start of my career.
Managing people is more of a craft. Approach it the way you would learn to build furniture as a hobby. Gradually over time through practice and experimentation acquiring tools for your shop along the way.
Is it your first time? If so, you would probably have similar growing pains MBA or not. Being a business conversant technical CTO is the kind of thing you only get good at by doing it.