Actually, I know multiple people who are in residency or have been in practice and are doing an MBA. Many do executive MBAs (EMBAs) and others do the full-time. The latter are usually people who want to pivot sectors, so like going from medicine to consulting or investment banking.
It is also not the case that anybody goes to business school for the knowledge. Getting an MBA is about meeting people and networking. School is secondary. Most things taught in an MBA won't help anyone, including physicians but also the anybody else in the program, in their jobs in terms of technical skills. What it will do is get them the job in the first place. The rest of the learning is on the job.
There are considerations other than those mentioned with respect to timing. Cost is a big one. If you have loans, taking out an MBA is even more loans and all that is accumulating interest while you're in school. You're also delaying your attending income by a year. In the grander scheme of things it probably won't matter, but it's a judgement call.
Also, I'll mention that in general, running a private practice doesn't require an MBA. Remember that most small business owners in the country, in all sectors, don't have MBAs. If you're talking about running a group practice or wanting to run large health systems, then an MBA may be useful. The MBA 1) allows you to network with people who can get you those jobs in management either now or later on (the earlier you do the MBA, the less likely you'll be able to capitalize on those connections later on as they become more distant) and 2) teaches you how to lead effectively (also known as soft skills).