In my opinion, the free market sucks hard at providing medicine and I'm confident the government could do it better but if I'm wrong, then nobody would go to the government doctor, they'd just keep paying to see their private doctor.
And that would be fine with me. However why haven't Obama done his healthcare reform this way? By setting up a government-owned doctor? I suspect it's because such a system would be very expensive; more expensive than private healthcare. It's easy to see why: nobody would be interested in working efficiently. Everyone would be working to earn maximum money while doing the minimum work. In private sector you have the patient who monitors your work by his wallet. In public sector the patient does not pay; and those who do pay are far, far away, and all that they ask for is mere paperwork. That is easy to provide without actually helping people. This is what happened in USSR. Doctors went through the motions of seeing patients. Not all of them cared; but those who saw more patients per day were rewarded. Those who saw fewer patients (and spent more time with each) were called "underperforming."
Socialized services tend to spread over time because they are inexpensive, high quality, and people like them. Imagine that.
That's not now things work. I was there. Were you? If you haven't seen firsthand how destructive socialism is, please ask those who were there. What you are saying is just a nice dream. Hint: nobody likes socialized services because they are always of low quality. A popular Soviet joke says: "You can use free healthcare, but only if the outcome does not matter." People used to bribe doctors to get a treatment that is just a little better than what is allocated to everyone else. There was no reward for doing a good job - the feedback loop was too long, and the salary matrix was cast in stone. It's not so when the customer pays you on the spot and then recommends you to his friends.
Your anecdote about USSR medicine really points to a lack of democracy not some problem with socialism.
What has democracy to do with socialized services? The UK is democratic, officially, but NHS is just as deadly as Minzdrav was in USSR. You can elect one representative or another, but if there is a fixed sum of money to treat everyone, everyone gets treated equally. Since there is never enough money in public coffers to treat everyone well, everyone is treated poorly. Usually the plank is set to just keep people alive and functional, for some period of time. Losing ten or twenty teeth is not an impediment - you still can sweep the street or work the machine.
It is truly sad how much societies love to step into the same pile of $hit, over and over again. Nobody cares about *actual* processes and causes that resulted in failure just a decade ago. First, everyone is saying "no, they haven't done it right, but we will." Then the history is forgotten, and old devils are reincarnated as saints. (Che Guevara was a psychotic killer, for example.) Now Socialism and Communism are in favor again; all the rivers of blood that were spilled due to them are conveniently forgotten, and people are $pleasing_themselves with ideas that were tried and found to be absurdly untrue. Eventually they get their wish... and then they will regret it. Again.
Does this mean that we cannot have a plentiful and free society? Of course not. But that society will not be Socialism. It was *a crime* to have a private business under Socialism. Do you want to be imprisoned for making a small profit on repairing computers for other people? That's what will happen. Socialism suppresses private enterprise because it cannot compete with it, other than by force of law. The end of socialism was written on the wall when Gorbachev allowed private enterprise. Government-owned sector could not compete, and did not want to. It died on its feet, standing still and not caring. Socialism is a non-free society by definition, and you cannot build anything good on that foundation. You must start with the idea that all people are free - free to act or not to act, free to associate with others or to stay alone, free to work or not to work. Only that foundation will allow you to create a happy society. Will that society care for a disabled person? Most likely. Will that society care for a person who refuses to work? Perhaps, on the most basic level. And here comes the inequality part. People cannot be treated equally if they themselves are not equal. As matter of fact, even the slogan of Socialism says "From everyone according to his abilities, to everyone according to his work."