Singapore has user-fees too, yet it also have universal health care. So while I agree that the US system is a shambles and certainly not to be emulated, we can't say it's all because of user-fees otherwise the Singaporean system wouldn't be so great either. I think the problem in the US has to do with a combination of user-fees and other things.
Singapore is a tiny nation (a city state) and a dictatorships besides, Singapore is hardly relevant to any discussion about health care. If they ever have any problems such as cost overrun, shortage of doctors etc., all they have to do is pass a draconian law (raising taxes, or mandating that so many graduates must go to medical schools etc.) to solve the problem.
Like China's one child policy. When you compare a democracy to a dictatorship, you are comparing apples with oranges.