Politics Revisited (Finally)

I apologize for making few posts lately, but I have recently gotten worked over the idea of universal healthcare, so I decided to write about it.
In America, our doctors have in their best interest the proper treatment and/or healing of their patients. If they do their jobs, we get better and they get money. This works very well, albeit I will admit that not everyone can afford treatment(I will talk about this later). Because doctors want to get paid and patients are willing to balance quality with price, we have a competitve market for the best and the cheapest doctors(although they are not necessarily different entities). Despite this, some countries have decided to insitute universal healthcare, which means that all the doctors get paid virtually the same amount, and the one who pays them is the government through our tax dollars. Now, as you might expect, when the government is paying bills, they always look for ways to cut costs. They can't cut the doctors' pay, but they can cut down on the number of doctors and the number of patients treated. This means that it is in the governments best interest financially to tell you that you are not sick, or else that they have no room for you. There are plenty of examples of people in Canada(which has a form of socialized medicine) getting flown into the US for simple medical procedures that would have been performed if the doctors were able to compete by means of multiple health insurance companies. We don't want this in the US. It will not help us at all.
Now, regarding those who cannot pay. I propose that we use universally available healthcare. If we offer healthcare to those who are not able to pay for it as it is, then we can solve the real issue in this problem. Now to address the problem I pointed out before: The government in this case would not be able to offer horrible coverage to those who use the healthcare, because the people would see the coverage given by other companies and demand something similar, in the same way a union works.
Personally, I think this is a good happy medium that will fix our problem.