You're describing one system, not all systems that use single-payer. I can't remember if it was Germany or a Nordic nation that was profiled, but for them you just showed up for GP visits.
Saying, in effect, that some countries do it right and some don't really makes my point. If you implement this in a nation of 300,000,000 people where moving to another country is kind of a very big deal, you're hosed if your country doesn't do it right.
Now consider that I live in a country that periodically shuts its government down because they can't even pass a budget.