I think people confuse Federally provided universal healthcare and single-payer healthcare. In a simple single-payer system, hospitals, Physicians, labs, etc can still be private and for-profit, single payer reduces overhead costs significantly while allowing for more competition and more choice for consumers. Single payer has been endorsed by the ANA, NNU, and the AMA among others.
In our current system, most of the money meant for healthcare gets filtered through insurance systems. The service insurance systems provide is to collect bills, store that money, and then pay bills. A fairly simple service for which they take between 14% and more than 20% of the money that gets filtered through them. Based on non-profit payer systems we know that the cost of providing this service is between 1.5% and 2%, meaning they skim an additional 12-20% for profit. Single payer would only keep this 12-20% within the system (most likely as cost savings to consumers) and would not your options in what Doctor or Hospital you go to, in fact it would do away with the "preferred provider" lists that most insurance customers are currently limited to that limit choice in not only Physicians, but also hospitals, labs, etc.
The company I work recently put a smaller competitor out of business by cutting a deal with a couple local plan providers to discourage those who use that plan from using that lab's services, effectively limiting competition and choice for patients in my area, something that wouldn't be possible in a single payer system.