In Holland we had a great mix of private & public health cover using the same facilities so it can be done.
In South Wiltshire the NHS and rivate co-exist quite well. We have Salisbury & District NHS Hospital that isn't bad, does acute stuff really well. About a mile away is New Hall Hospital. Privately owned it does elective surgery, the non acute stuff. Same surgical teams. Abut 50% of their work is as a sub-contractor to the NHS, about 50% funded by "top up" insurance.
So, if you need a hip replaced the NHS will do it, when they get round to it. You will stay in the private hospital, where they ask if you want the NHS menu or the private one. The private one is £15.00 a day extra, but the food is good.
But if you want a single room, not a room with 3 beds, and if you want to set the date to suite you, eat well and have a glass of wine with your evening meal, then either you or your insurance company pays.
The other difference is that on NHS tarif you get a hip from a smal list that "NICE has approved. Privately the surgeon can chose what fits your needs from a much woder range. Caroline had both hips done privately when she was in her late 40s early 50s. The NHS hip option would have lasted 10 to 15 years, then need replacing. The private one cost about 10 x the NHS approved one, but is designed to last 30 plus years. They probably won't need replacing.
The NHS doesn't look at anything as a business would. In Caroline's case a simple discounted cash flow calculation whould have shown that over 40 years the expensive option actually is the lowest all up cost. As someone said, they know the cost of everything and the value of nothing.
The problem is in the title. National Health Service. It isn't a Service, it is a badly run "nationalised industry", like BL. In my view, as in Germany, all points of delivery should be run as private businesses, regulated and conforming to an audited quality management system, but able to fail.
The Government Insurance simply pays the agreed tariff.
Can you imagin a "National Garage Service" with no real choice where you service your Mog, or who does it?