The vast majority of private work is carried out in private hospitals such as the Spire hospitals. There are other groups too. The NHS uses private hospitals for surgeries and other treatments, and so it would be foolish for any government to bite the hand that feeds it, so to speak. The government needs the facilities that private hospitals offer.
As for doctors working in the private sector, remember it is only consultants and well qualified doctors who can do so, and consultants work under contract to the government, and so they work whatever hours they negotiate and are only paid for those hours. So some might work two or three days a week (particularly parents who still have young children) whilst others may work full time, and indeed work overtime. Consultants can be called into the hospital at any time when they are off duty to see emergency, complex cases. This happened to me. The consultant came in to see me on a Sunday lunch time. He was in the middle of lunch at someone’s house, but left the lunch and came to the hospital.
I think many misunderstandings occur because people don’t actually know or understand how doctors contracts work. Please remember, we do not own medics. They are free to negotiate and sell their skills to whoever they wish, just as other occupations do.