Most of Europe has a similar model to what you've described: some form of mandatory state-backed insurance combined with a mix of private and public healthcare providers. In many countries, however, the biggest hospitals in Europe are owned by the government.
Unfortunately the healthcare systems are in the process of collapsing across the UK and the majority of the continent too.
My pet theory is I don't think it's actually anything to do with the overall funding model. I think it's to do with our inability to adapt to an increasingly elderly population. People's kids here are scattered around the country, often many hours of travel away, living in small apartments, and can't easily look after their elderly relatives in a way that's much more common in East Asia. As a consequence, we are offload that responsibility onto the healthcare system, which treats them as patients with medical issues, when often they are just old people with broadly normal age-related disease. Our systems were never designed to be capable of handling millions of elderly people, and it's not an efficient way of providing the required care, so it's falling apart.
This is an interesting theory. I just came back from Thailand where I needed to make an ER visit for a grand total of $89 USD, including the price of three prescriptions. I was touring with one of the natives who made a comment around the lines of, "We don't abandon our elderly parents here like you Westerners do." This was in a conversation around their multi-generation households.
It's not necessarily a correlation, but your comment reminded me of the conversation.
Unfortunately the healthcare systems are in the process of collapsing across the UK and the majority of the continent too.
My pet theory is I don't think it's actually anything to do with the overall funding model. I think it's to do with our inability to adapt to an increasingly elderly population. People's kids here are scattered around the country, often many hours of travel away, living in small apartments, and can't easily look after their elderly relatives in a way that's much more common in East Asia. As a consequence, we are offload that responsibility onto the healthcare system, which treats them as patients with medical issues, when often they are just old people with broadly normal age-related disease. Our systems were never designed to be capable of handling millions of elderly people, and it's not an efficient way of providing the required care, so it's falling apart.