'Even if you have private insurance in the UK, your health care is affected by the NHS. You get a better standard of care (if only measured in a private room versus a crowded co-ed public ward), but availability of doctors and equipment is still impacted by the system.'
Not so. When I use my private healthcare, I go to a private hospital with private doctors and nurses - it's nowhere near an NHS hospital.
There are 100% private hospitals and doctors, but there are also private sections of NHS hospitals, and most medical workers divide their time between public and private commitments. They overlap and effect each other.