You really are something else. Did it never occur to you that "free healthcare for all" and "free education for all" are somewhat undermined by the concurrent availability of private education and private healthcare?
Here's one for you: My ex-father-in-law was a neuro-surgeon at Hope Hospital in Manchester. He told me he could earn 95% of his NHS wage whilst only working 60% of the time doing NHS work, the rest of it the lucrative private stuff. Worse than that, he was contractually obliged - even though he absolutely detested the notion - to work 10% of his time in private consultancy work. He did the minumum private work; his colleagues did the maximum.
If that isn't private enterprise undermining equality by selling "queue jumping" to the better off, I don't know what is.