> with the caveat that there is NO private pay option

I don't think there is a society in the world that ever had that successfully executed in practice. I live in Norway, and even if the public heath system is probably as good as it gets here, I often use private providers for non-urgent things simply because waiting time can be up to six month otherwise.

And even if you somehow manage this on paper, you'll be very surprised on how markets work. They don't need legal recognition. They don't even need money. See: USSR. Health care was public and medics weren't even a particularly attractive profession because there were too much of them already. Still if you wanted to survive an appointment you'd better have a package of some deficit wares ready.

What denies public healthcare money isn't "wealthy trying to cut public spending down". It's bureaucrats wanting their share of the pork barrel.