Pretty much everything run by the government involves high prices and bad service.
So I don't see why a government run health care would be any better.
That being said, Single payer is not "government run" it's just paid for by the government.
We don't live in a post scarcity society, everything is scarce, including health care. So if everyone had the exact same access to the system, there would be more people then the system can handle and some people would not get the care they need. So not much different than the way it is now. And unless you ban private healthcare related transactions, the people with means and money will get the health care before the poor.
What I believe would happen would be that everyone starts paying more taxes, taking money directly out of their pockets. Now your average person has less money to spend on things that could potentially make them healthier (Gym membership, a new bike, better food, etc). All the while the rich would still get better care, and simple things like medication or visits would be rationed (due to scarcity). Only now instead of the health care industry being blames, the government would be blamed.
In the end, people need to realize that there simply aren't enough Doctors, Hospitals, Nurses, Beds, Pills, , , , etc. for every single American to get what they need. So long as that is the case it doesn't matter who is in charge or who pays for it. Everyone will not get what they need.
IMO the only real solution is to completely open up the market.
If you want the government involved let them pass a bill that says something like "Any doctor who relocates to the US and practices for 10 years will get a federal grant of $?????.??" and let them build hospitals and subsidize drugs, treatment facilities, medical equipment, etc.
If hospitals were a dime a dozen and everyone of them was fully stocked, then prices would go down and access would go up.
Side story.
My sister is a nurse. When she applied to the program she got an e-mail saying "thank you to all 2,462 applicants, unfortunately we can only accept 200"
Meaning that a few thousand people wanted to be nurses, but for some reason (The AMA and the government) limit the amount of new nurses in the field.
And the shitty part is the limitation is by graduating class, not an overall standard. Meaning that one year the bottom end of the graduating class could be lower than the upper end of the people who were cut the year before. It has nothing to do with "quality" and has everything to do with artificial scarcity.
Imagine for a second that there was a "Auto Mechanics Association" and they only allowed 1,000 new auto mechanics to be certified a year. Now say that they are so powerful that they lobbied for laws that make it Illegal for anyone who is not certified to work on a car, and illegal for you to work on your own car. What would happen to access and prices? The exact same thing is happening in the Medical field. And that is the problem. We don't need "free" health care. We need more doctors, more hospitals, more nurses, more medicines, more competition, and more of everything. The closer we get to post-scarcity the cheaper everything will get and the more access we will all have. But with the AMA, lobbyists, and horrible government regulations we will never get there and people are literally dying because of this.