Um, I pick my own docs, have never had any serious delays in getting treatment, nor has anyone else I know who's had more serious issues (from knee surgeries to cancer). More to the point, a cancer patient I knew changed docs several times, and had basically the best care the world has to offer, all under the U.S. system. And they weren't bill gates, they were average middle class folks.
My understanding is if you have one of the major insurers in the U.S., most doctors will accept your insurance and you can pretty much go anywhere you want. Obviously better docs have longer waiting lists but I fail to see how that would change under any system short of cloning.
As far as medical care goes, in the U.S. if you have private insurance there really is no better care you can get anywhere in the world, unless you are a billionaire and don't have to rely on insurance. It's the uninsured that are having issues in the U.S., and last I heard 95% of voters have insurance which is why, generally publically funded health care here is a non-starter.