First, "Single-payer" is a myth. France, Canada, Britian, Italy, Sweden, don't have it. They have a system where the state provides a minimum that the state defines as sufficient. However, there is also insurance and private doctors for people who can afford it.
There is a term for it over there. It is called "Going private." This is not a term I made up, but part of their medical lingo.
THe main problem with a true "single payer," is it is also a "single decider." The entity that pays the bills decides what you can get. THis means that if there is a new treatment that is "experimental," it may be a better medicine, but it is expensive the first years while tens of millions of dollars in research are recouped.
However, on a cost basis it is therefore not worth it, so the state won't pay. The problem then becomes fewer new medicines are developed because the cost is too high.
Obama himself spoke about "smarter spending." That necessarily means someone has to decide what is smarter, and if you have single-payer, in a true system, it is a committee in D.C. or Ottawa or London.
If you want something the committee hasn't approved, tough luck under single payer because no one else can pay for it.
Again, that is why the U.K., Canada, and every other "Single-payer," have private insurance for those that can afford it