Main problem about this topic in the US is the twisted belief that everything public = communism.
Followed by a strong inclination to see everything black or white. Different shades of gray do not exist in the US.
You don't have to have a 100% public health system. We have a lot of problems in my country (Italy) but we've recently been ranked #2 for the best health system in the world, behind France. Over here healthcare is completely public with minimal fixed charges for everything (never seen anything above 40 euros, and I'm talking about stuff like 10 sessions of physiotherapy + specialistic visit). Over 60 years old (or 65, not sure) and below XX (kids, I don't remember the age), everything's free.
But you are NOT forced to go public. There are plenty of private clinics, wich have to be approved by the public system, but are 100% autonomous. Those are expensive, obviously (up to 200 euros for a simple orthopedic visit), but usually things are a lot faster.
So basically it's a choice between "almost free but somewhat slow" or "faster but you pay".
Obviously emergencies don't have to wait at all, even with a public system.
I'm talking from personal experience here. A year ago I've totally fucked up my right elbow in a horse accident. Exposited dislocation with multiple broken ligaments, and the wound was full of sand and small pebbles to boot. Urgent surgery was necessary, obviously, and I didn't pay a cent. No questions asked, nothing about my insurance, nothing. You fuck yourself, you go to the hospital, they fix you.
In the US this thing would've costed something between 15,000 and 20,000 dollars. And I'm talking about the surgery alone, not the following 80 sessions of therapy and 20 of ultrasounds. No idea about those, but I've paid 36 euros for each 10 sessions. And guess what? My insurance refunded those.
I'm far from done with this elbow (joint-wise, the thing was basically amputated) and I need more surgery. This time I'm going private to speed things up and my insurance is going to pay everything. But it's my choice. I could easily continue with the doctor who fixed me the first time. A bit slower, but free.
Now my country is totally fucked up in other aspects, but I'm pretty darn proud of our healthcare.
What you have now in the US is a disgrace. Any modern western country should be ashamed of that monstrosity, even more so a great country like the US. I've never managed to understand what's so HORRIBLY wrong about a public healthcare. I can't even imagine the consequences of a hypotetical total privatization here in Italy or any other western european country.