What do you pay for health insurance? How many people in the lower levels of society could afford to pay what you pay for healthcare? Do you have an employer that provides you with healthcare benefits?
I agree that healthcare in many countries could be greatly improved, but at least in many of those systems you could just walk into an hospital to start the process.
The reality is that you're in a privileged position, and it may not seem like it to you but there are a huge number of people out there for whom your situation would be effectively a bankruptcy trigger. And that's even assuming that their company lets them off work long enough to get treatment and recovery, which, in at-will states, doesn't seem like a thing that's likely to happen for a lot of the working-class.
For rich people/folks with wonderful insurances the other countries with public healthcare have private care in private hospitals, and it's of the same class — with one-on-one nurses and good food.
After reading on the experiences of people from European countries and Canada in the 'OMG I had a valve replaced' forum (yes, there's a site just for that -- valvereplacement.org), the level of care that I received was significantly better than what is typical of public health systems.
Oh, and last week I needed to see the doc for a sore throat, and I was in the same day. Queues are a regular thing in public systems.