If a service member started his own business and was able to meet all of their service requirements, would there be a problem?

Yes, you can do this. You will have to look into your particular service regulation, to take outside employment you typically need command permission. Also look at the definition of employment in the regulation, being your own boss may make the requirement not apply.

Believe you can do this, it requires permission from the CO, though. There is a YouTuber named Nick Bare. I believe he was an army 2nd LT. and made quite a lucrative business from selling fitness products and creating his own gym. I know he mentioned the process of doing so whilst in the army, but I dont know the specifics. May be worth checking out ?

Don't really know. Depends a lot on the command, the duty that the member is assigned to, their "other duties" (watch standing etc)... Back in my days, you could put in a special request chit just to ensure you were on the right side of the Command.