I recommend those for single payer (or a Swiss type system) just admit these systems ARE socialism. E.g. for such a system to work, medical providers (physicians, hospitals, therapist, etc...), would have to be subject to price controls (I.e. a medical fee schedule that sets reimbursements like with Medicare). This has been the case in the Workers Comp system since CA (yea, that state again), introduced a fixed priced fee schedule in 1969. The WC market place is a socialist system.
I don't wish to get into a discussion about what is socialism, but even those that wouldn't define the above as socialism have to admit these closed-systems are NOT capitalism and as you note, that would be the main objection by conservatives, independents and maybe even folks like Pelosi (based on her put down of a Sanders supporters where the first thing that came out of her mouth was 'we are capitalist').
Workers Comp is fairly small when compared to health care for everyone. Assume that CA is like most states, there is medical treatment for a while, then a cash settlement and then case is closed. Also, the care is provided due to an accident or similar event, not routine care.
If we want to allude to socialist health care, more accurate would be VA, military retirees and dependents and possibly health care provided for government employees. Veterans and military retirees earned their care, but they do not pay for it. Government employees may pay something, but fairly small compared to overall costs.
Of course, Medicaid is pure "socialism."