I can only speak from experience with regards to the United States, but in acute care, they don't have to be doctors everywhere, and in some places they cannot be doctors per statute. I did business brokerage in the medical sector, although there are ways around just about everything. In veterinary industry in some places non-veterinarians are not allowed to, but again, there are ways around this in terms of organizational structure making it a moot point.
In acute care there is normally a position called a medical director. Very often the CEO is someone with experience in the medical sector, like a nurse, but it doesn't have to be.
Medicine is definitely a business. Hospital executives get paid a lot of money - FYI. Also, non-profit doesn't mean poor. There is an issue in general in the United States were non-profits enrich their executives because less people are watching the bottom line (they're not trying to maximize profit). In many ways, its worse that they are nonprofit. If they are a non-profit though, you should be able to look up disclosures about what they they paid their executives.
I have a lot of experience with this and I have also analyzed a lot of medical firms, including consolidators.
A good resource for information on hospitals in general: https://www.beckershospitalreview.com/