Same here. We have private insurance on top of it (extra insurance). But all it actually gets you usually are reduced waiting times (when time is not of the essence), and sometimes - higher profile physicians. (Which arent always 'better'.) So it really can be argued, that it isn't needed at all.
What socializing healthcare actually does is set a fixed price on all default procedures, and helps to drive down cost of medicaments, because you have large bodies (ensurence companies), that represent close to everybody - going into price negotiations with the pharma industry.
It doesnt mean, that Pharma cant make any money - they still offer up some of the best paying jobs - if you are willing to drive around in the country and influence doctors out there. It just means, that one form of price gouging is eliminated.
There is also the part of "yes socializing also means, that you are paying for the poorest" - but then hopefully that part of the demographic is so small - that the average tax payer, especially in your country with at least 1/3 more in per customer spending than the rest of the world, doesnt even notice it, and pays less over all.
To "our medical system is the best though" - the reply is usually - canadas isnt so much worse - and look up its graph in the link above.
A "two tier" class system will always exist in some form I bet. Its just, that the market alone in your instance doesnt fully work. (Because of basically market consolidation (monopolies)).
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In our country we actually have a multi tier system - where farmers for example get a little "better" (more well paying (for what its costing them)) insurance, because if they have health problems they are more at risk of loosing their livelihoods. That of course isnt very openly advertised, but anyone can look it up.
In general - our medical insurance system is solidarized (/socialized).
In terms of getting good physicians - all university level clinics are at least partly state sponsored - and are open to the public. Thats how you ensure that everyone can get high level care regardless of the issue. And yes, if it is necessary - it is free. You are just paying your base rates, and maybe a small fee everytime you visit a clinic instead of a local doctor (to disincentivize people always running to clinics first..
).