Everywhere I've lived (except the US) has had a totally free public system, and every time I've still paid for private insurance. I'm not rich by any means, and I don't have any particularly special requirements. I'm not even looking for better doctors or machines, since they are mostly shared or equivalent between public and private care.
The only reason it's worth it for me to spend money on private insurance is waiting times. On the private system, I can see a consultant in one month instead of six, or get a surgery in three months instead of three years. This is all for non-emergency care of course.
There are lots of elderly people waiting on operations like hip replacement on the public system, and it's expected that some will die before their turn comes up. It's a compromise that's built into the way the government operates their national health system. It not obviously the wrong way to do things, since carrying out all these operations would be very expensive and have limited benefit in terms of quality and quantity of life. It also sucks for the people waiting, especially since others who can afford to skip the line by paying (usually through private insurance) may even be seen by the same surgeon in the same OR.
I don't mean to argue for or against this system, but just to point out that people often die waiting for (non-emergency) care in at least several large public healthcare systems in highly-developed countries in Europe.