I think you've contradicted yourself there.
Many other rich countries don't accept the degree of poverty that is acceptable in the US - the countries of Europe, for instance, and Scandinavia.
What people are talking about is a shared European moral sensibility or theory of a just society that developed from the 19th century on, broadly labeled socialism, with a spectrum of political groups that are mainly Christian Democratic or Social Democratic in philosophy. Virtually all mainstream European political parties are to the left of all American political parties.
There is a recent resurgence of nationalism in some quarters but the majority of voters all throughout Europe are happy to cast a vote for parties that support common European goals and a common European idea of what a society should do for its citizens. No such consensus exists in the US, where a big chunk of trump support comes from the desire to put an end to the administrative functions of government. Hence the chainsaw circus, complete with clowns.
There are bedrock assumptions that the majority of Europeans have - that science will inform health policy, that science will inform school curriculums, that welfare is a good and useful thing, that healthcare is a human right, that are shared by some Americans, with the rest inclined to shriek in horror at the word 'socialism'.
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Yes, having specific doctors and hospitals covered by certain insurance policies is called a preferred provider organisation or PPO. The alternative is an HMO, where primary care doctors act as gatekeepers, deciding who should be referred for specialist treatment.
The NHS is basically an HMO. You are restricted in who you can see and where, and the quality of care you receive depends entirely on where you live.The NHS also rations care via refusal of GPs to refer, and slow uptake of treatments like Tamiflu or universal GBS testing for pregnant women. Unlike the US, litigation isn't always used as a means of making failing trusts straighten up and fly right.