You're exactly right: That is most likely how America would implement socialized medicine. It's also the reason you're in this mess in the first place.
I live in Ontario (Canada). We have socialized medicine. I go the doctor of my choice and OHIP pays for it. I go to whatever hospital I need to, get whatever needs doing done and OHIP pays for it. I can have tests done at whatever lab is convenient for me and OHIP pays for it.
Are our waiting rooms full? Yes, for two reasons: Free systems are abused and people are stupid. Instead of using the second hour they're sitting in emergency to ponder whether or not they really need to be there, they go off on half-cocked rants and diatribes that Americans misunderstand and use to fuel the anti-socialized camps. Notice how neither reason was government inefficiency?
If I have seen farther than others, it is because I was standing on the shoulders of giants. -- Isaac Newton