I think you are moving the goal posts again to fit your argument.
By EVERY metric when Bernie and self-proclaimed progressives say “single payer” (and they also say Medicaid for all) they are explicitly saying GOVT run medical health care system- period.
No reasonable person hears them say those two phrases and infers some other elements to the program
Which again points to my critique in previous communication about buzzwords with lack of detail. If you don’t provide detail you’d better be precise with your terminology