The slowness you describe has not been my experience in many European countries. It really depends on how the healthcare system is designed to work.
I was pretty surprised when I went to China and I could just go to a doctor's office without an appointment, stand in a queue, stick out my tongue, get a diagnosis within ten seconds, and be sent out with a handful of random colorful pills wrapped in paper. Yes, this was, in fact, a real doctor's office, and I did take the pills. I have no idea what they were or did, if anything at all.
I also did enjoy the phone-ordered house visits from nurses in Indonesia when I got Bali belly, although that is probably a result of low labor cost more than anything else. And the poop tests I received a few hours later weren't all that helpful. Oh, I have "amoebas," that's very useful, thanks.
All told, I'm happy I usually live in Western Europe. Relatively expensive, but high-quality, fast healthcare without the constant risk of going bankrupt when something unfortunate happens.