We know it works because we have examples. It is quite frustrating to see you not even try to go and get those examples. It only takes a simple google search.

Here are three good ones: the UK, Japan and Canada. All different systems, all providing medical care for much less cost than we do in the US.

Additionally, your reasoning is flawed at best. The government can legislate that providers can only charge $x for y service, which private actors cannot. 300 million+ people is a very large market to leave just because you are slipping on profits. There are many other levers a single payer who is the government can pull as well.