We've got a system that is a bit in between.
Everyone is obliged to get a basic health insurance, but, can choose what insurance company to use, and with how much own risk per year. On top of the basic insurance, extras can be insured. For those who have a really low income, a big chunk of the costs of insurance are refunded by the taxes. Because of the choice of insurance company, there is competition. Insurance companies also help out in finding where to get treated the quickest. Some insurance companies offer that service also to people who are insured elsewhere. The way to get new clients. This competition does make waiting lists a lot shorter.
The obligation makes it a bit like the NHS. The choice of insurance company, and the option of getting extras insured, makes it a private insurance.
I have my basic health, and some extras insured. That means that physiotherapy is covered for more than 10 times a year. My dental costs are also insured, to a certain maximum. So far, I have not exceeded that.
For some things there are long waiting-lists here. Urgent things will be done urgently though.
I was born in the bedroom of my parents btw. The GP and a maternity nurse attending. A Dutch home delivery.