The person doing the assigning is normally the patient himself, online. If he wants. His choices often include private clinics, if they can offer the service at a competitive cost. Of course, it can be more inconvenient for birth if you chose to live in a very remote and sparsly populated place. But in general, you should not base your opinions on Norways system on "Lilyhammer" any more than you should base your opinions on the US on "Married with children"
Well, the issue with single payer (there is no such thing as "socialized health care" outside of american opinions. Its like the Swedish lesbian town.) is a bit more complicated than that. Through the system, decisions are made by doctors on medical grounds rather than bureaucrats basing it on on whats profitable. So access is set by triage. The more serious your condition is, the faster you are served. So you trade faster service for serious things for slower service for minor ailments. And the papers write about the conditions in between. Minor ailments always outnumber serious conditions too.
Beyond that, most single payer countries do not compare badly to the US in terms of overall waiting times. Canada, which is where a lot of Americans get their impressions, is an unusually poor performer in this area. Of course the US stats are distorted by the people who struggle for any access beyond the emergency room.
In general, the prevailing opinion in Public Health is that when you look at access in general, which includes both waiting lists, people with no access at all, and people who hold off seeing the doctor for financial reasons, the UHC systems are the more effective.
Government taking over healthcare is another Swedish Lesbian Town. Also, its not neccessarily true that you'll lack for the most cutting edge solutions. Single payer healthcare have an enormous bargaining power, and in some cases just buy treatment where the best treatment can be found. It can be cheaper than to maintain specialist skills for rare conditions for smaller countries.
Overall, the US does well in the "cutting edge" stakes, but the US system is open to anyone who can pay.