Neither will developed-world single-payer systems, in virtually all cases. It sounds like you're referencing the Alfie Evans case, which a) was unusual enough to receive international headlines and b) involved a terminal, untreatable minor incapable of making their own decisions.

Hell, even in the UK, you can opt for private treatment - they've got a dual private/public system, as do many single-payer systems.

Completely untrue in the UK. We have private healthcare alongside our NHS system. You can pay for it through insurance, or you can sell your house and pay for it in cash.

Doctors proposed a plan. The parents disagreed with that plan. The doctor's had to go to court to get permission to go ahead.

Contrast this with the US system. Doctors propose a plan and if the parents disagree it is them who have to go to court to stop the doctors.