It's worth noting that the modern US usage of the term 'socialized medicine' was invented, amplified and promoted by a pr firm for the for-profit medical insurance industry. https://en.wikipedia.org/wiki/Socialized_medicine#History_of...

No surprise really, the word "socialism" has been used to attack any pro consumer / pro employee policies since the first red scare 100 years ago. Indeed up to the '70s, even civil rights was painted as "commie".

Compared to dying because treatment a private health insurer pays for would simply be denied, as other payers are prohibited.

Why would you want to make medicare supplemental coverage illegal, or forced to be completely separate, which is a characteristic of single-payer health care (as opposed to multiple payer universal health care coverage)?

Supplementary coverage is perfectly legal in the UK, I don't know why people think it isn't. Maybe there's a disinformation campaign.

AFAIK the UK has optional supplemental health care, and Canada has optional supplemental plans for things not covered by the public plan.

That is not how public healthcare works in Canada at all.