Socialized healthcare systems often have room for a private sector, so you can have both.

You can also go to other doctors in a socialized one, if their treatment (style) does not fit your cause.

Also, in my experience there's lots of choice in most socialized health care systems.

I had a gastroscopy in Austria, the Doctor asked me if I'd like a general anesthesia or rather a local, nerve blocker - mostly to avoid the gag reflex.

He said that it may not be as pleasant without the anesthesia, but I still opted for it and am really glad I did. While it really was not pleasant, I was up and going again in about half an hour.