P.S.: Thinking about this is making me recognise how wonderful is to have free public health service and how we usually don't see the value in those things we have always taken for granted.
It's not free. In this case it appears you don't see the cost.
Of course it is being paid in advance in order to be "free" when you need it but, there is something very wrong with the alternative system:
Look at the prices of medical procedures in the US. One would suppose that capitalism would manage to reduce prices due to competition as it does in most other sectors. But in this particular case it is not happening at all. How do you reconcile US not having universal health care and still having the biggest per person spend? It just doesn't make any sense.
I don't believe in socialism, but in this particular case I have to acknowledge universal health care works way better overall.
Of course I find completely ridiculous that universal health care also covers foreigners (and even ilegal migrants) here whose respective nations don't reciprocate when we go there. That's completely stupid and plain wrong from an economical point of view (I will leave ethical concerns out of this) and it is well known there are many people exploiting this.
I don't know nearly enough about the US health system to answer this, but I will say that part of the problem is that the US is not a capitalist country. It is as socialist as everywhere else in the west, it just has a slightly different form.
Another potential issue is the sue culture. It's so out of hand that we make fun of it in the rest of the world. What's a medic to do when he risks getting sued every time he treats a patient? Well he obviously has to charge enough that he can afford that risk.
I have stayed at a number of hotels in thailand with pools over the past months, and they all have the same sign with the same clause: There is no lifeguard, and you use the pool at your own risk.
Remove the welfare state - all of it, rip it out by the roots - and change the legal system so people are forced to accept a higher degree of personal responsibility for their own actions, and prices might eventually come down. But it's a decades long process, even if done right, which it won't be. Basically the whole country is fucked.
Yes, I can understand the "sue culture" has a big impact on prices. You really need to set crazy prices to be able to cushion fines that can even reach the hundreds of millions here and there.
Removing the welfare state? I donno... Maybe that's too much. Do you know what kind of things can do people that doesn't have enough to feed themselves? I don't think the welfare state was created just because of making sure everyone is well fed for... charity... it is also for safety of the rest.
Also, the health system is very complex... If it were fully private and only for a few that can pay for it there would be no economy of scale and probably the cost would skyrocket. As much as I do believe in capitalism, until I see a detailed and well thought plan of how it could be implemented and what it costs would be I wouldn't vouch for it.
Maybe Amazon has a plan for that... if so I would like to study it.
In the meantime, I prefer euro universal health care even if there a lot a changes I would do... The first one being charging service costs to anyone whose country doesn't reciprocate the same.