When you have private insurance companies you necessarily have the private insurance company bureaucrats making health care decisions for people. 

"At least with an insurance company, the limit of it's authority is to say "No we won't pay for that" It doesn't have the legal authority to tell you you also can't pay for it yourself," is a red herring. 

Medicare patients in this country are allowed to see a doctor who doesn't accept Medicare if they are willing to pay for it themselves. In the UK, people are free get treatment outside of the NHS. 

The decision not to allow Little Charlie's parents to spend their own money to bring the kid here was made by a court, not by the health care system.

The claim is bogus.