I'm in 2 minds about this. Years ago, my MIL was sent by the NHS to a private clinic in France to have a hip replacement. Great Service, rehab in another centre so no bed hogging and all a complete success. At the same time, my mother had hers on the NHS in an NHS hospital. Lots of problems, poor decisions, infections, etc. 3 years later she had to have another hip replacement. More recently, a friend had a knee replacement with great success at a private hospital under the NHS. My mother had one within the NHS. The surgeon cracked her femur but refused to admit it even though it was clear on the X-ray. The whole thing was a nightmare.
My son has been sent to private hospitals to have spinal injections and the service has been brilliant. The one he had in an NHS hospital was done eventually but there were minor problems on the way.
If the private services are procured by the NHS but they remain free, then I don't see a problem if it keeps waiting lists down and the services are good. What I object to is services just being withdrawn completely. My mother went to have her hearing tested and they couldn't do it because they said she had blocked ears. The NHS no longer does this so she had to pay £80 to have it done privately. No problem but I do wonder how many low income people can afford this and then they are subject to deafness, isolation, etc.