Single quasi-private payor should be the only way to go. It should be out of the government but regulated. Blue Cross of the US, perhaps of every state, but the government is to fraught with possibilities for corruption (anyone need a senate seat??? (sorry, I'm from Illinois.))
It's illegal to have collective bargaining for physicians. It's OK that the private companies have private rates that are suprisingly similar but then with their own rules to confuse both providers and patients.
However, if we went to single payor, it would probably cause a full scale depression with displacement of about 500,000 jobs. Talk about a works project.