I am from Denmark, but married to a US citizen. We have a lot of opportunity to compare notes. While Danish doctors are often somewhat rude and will cheerfully refuse to give you a prescription for stuff you are sure you need, we would never see a case like my wife's uncle. He lost his leg because he didn't see a doctor about the pain, and his reason for not seeing the doctor was that he was worried the visit would not be covered by his insurance. When he finally went, it was too late, and they had to amputate. So it goes. Meanwhile, in Denmark the government is often imploring the citizens to see their doctor more often, to keep health costs down by spotting problems before they become expensive to treat.
Personally, I have received many, many treatments ranging from setting of broken limbs to specialist examinations for this and than, and every night I use a C-PAP machine, paid for and maintained by the socialized health system, but supplied by a private specialist. I can, in fact, choose any doctor I want as my GP, or just make an appointment or show up as a walk-in. The only practical limit is that in order to see a specialist, I need a referral from a GP. This has never been a problem for me.
Our system? Socialized with a private option, with an overflow to the private system if the public system is too tardy - again at no extra expense for the user. You can add a private insurance if you wish, and many people choose to do so for things such as dental, plastic surgery etc, but it's really not required to stay hale and taxable