Single payer government run health care that eliminates private insurance and the vague statement that "the government should make sure everyone has health care" are not the same thing.
It's
the fundamental question. You can poll on the broader subject with any number of broad or specific statements that are designed to suppress or expand support for the concept, and end up with anywhere from 35 to 65% of the public supporting the concept. The primary opposition against enacting single-payer
does not reside within the public at-large. Any time you provide specifics on a broader issue, public support shifts (and often erodes) - but not always for the reasons some people would like to assume.
One example would be to ask if people support "universal background checks" - 80 to 90% agree. Begin articulating each specific piece of what is required to implement it, and support begins to plunge markedly. A counter-example (where people support a concept more as they learn about it) is ACA: ask people if they like "Obamacare" and a narrow plurality or majority might say no. Ask them point-by-point if they support what constitutes "Obamacare", and it's an aggregate respectable majority in favor.
The common denominator is that the public doesn't know much at all and can be persuaded in either direction by whoever has the best narrative. This is the biggest reason why ACA was a flop, in my opinion: we lost the narrative and it cost Democrats far more than it was worth. At the end of the day, single-payer would follow the same trajectory: people would like the broader idea, grow to dislike whatever actually manifested as advertised, but
then would actually like the individual provisions when explained.