That is already happening. The NHS is paying the private sector to perform operations due to long wait times (it will take half a decade for the waiting list to go down)...but these are usually the same NHS doctors, working in the same hospitals...they are just being paid 3x the price, it is the worst of all worlds (doctors are an extremely powerful lobbying group in the UK, they have tightly controlled entry to the industry so you have doctors making base £150-200k part-time on the NHS, then topping up with another £200-300k with private sector work...it is chaos).

I understand no-one in the US wants to hear it...the UK system doesn't work (fully public health doesn't work, almost no countries in Europe with good healthcare have it, Australia is very similar to the UK, introduced private healthcare...their results are miles, miles, miles ahead).