Some Eastern European countries have a good system. You go to your GP, and if after examination they decide to refer you, you often see a specialist the same day. You are expected to wait in the waiting room for hours, but you see them regardless. People have insurance , which you register with your GP etc. Your employer pays for your health insurance unless you're self employed pay yourself or unemployed paid by system.
Children don't go to see any GP, they have paediatric GP surgeries.
You are always guaranteed an appointment. If you're are not happy to wait at GP surgery, you can pay for a appointment time slot.
Technically,the health care is a small fee you pay or your employer pays.