Countries with Universal healthcare (and they're not ALL single payer, by the way), most often control costs by constricting supply. Canada is the easiest example, and it does indeed keep their costs down a bit, but it makes it far less responsive and efficient as well. Plus, Canada isn't even completely universal or single-payer- citizens still have to buy either government or private insurance for their prescription drugs. Another way they limit costs is by combining facilities, placing caps on physicians fees, and closing hospitals. Ask a Canadian sometime what 'Summer Shutdown' is. If you're a patient waiting on surgery, you won't like it. But problems with cost cutting aren't relegated to Canada. If you're a couple in London who have had two miscarriages and you want to find out if you have fertility problems, tough luck. The UK healthcare system won't approve you for testing you until you've had four miscarriages. That's just the way it is. It's too expensive otherwise.
Now- Canada is probably the last of the "universal" systems we would want to emulate compared to France, England, etc. But those same cost cutting measures apply there as well. But Canada shares some of the same problems we would face with a universal system- namely geography. Centralized facilities those countries rely on for efficiency can only exist in a country where the entire population lives within a few hours drive of the center of the country. Centralized facilities are cheaper, and patients can reach care easily. They require fewer doctors, far lower costs, etc.
There's a problem here- we have 300+MM people spread over a continent. We'd need exponentially more facilities (and therefore doctors) to provide those services to our population with any semblance of accessibility. And it's not just proportional to the population difference either- we have so much GEOGRAPHY that delivering services to huge swaths of the population require a ton more facilities, distribution, etc. It- in very simple terms- will cost significantly more per capita to emulate what other countries do here. Otherwise, it could take days just to get a patient TO a specialized facility, or the cost of moving patients in a prudent time would be astronomical and would actually DECREASE Accessibility.
So it's not just a matter of dropping another system into our country and having it work the same. It can't.
But something has to give. The free market doesn't work with healthcare in our system. It just doesn't. The inequality of access is proof of that. We don't want to lose what our system does well- which is innovative and TIMELY care at our doctor's instruction, not the government's. What we need to find is a way to get the quality and efficiency of care we already provide to everyone who needs it rather than to those can afford insurance and the resulting copays. That has to start with some reforms that are intended to either stop the unmitigated growth of costs or even reverse it in some ways. Then we need to make that care accessible. That's why I say I don't know what the right solution is. It's not, in my mind, the system in place north of the border or in European countries. But it's also not what we have here. I have to believe that if you took politics out of it, we have some smart people who could come up with a new system that combines the best of both worlds. Whether that's single payer, government/private universal, or some other form of private, I can't say.
But every time we start trying to further bastardize insurance to do that, we make the problem worse. Unfortunately, the only thing our politicians seem to be able to do is focus on changes to insurance and claim they've fixed everything and walk away. Insurance, as it exists today, is a symptom of the problem, not the problem itself. And you don't fix a problem by treating the symptoms.