Like all these issues it's more nuanced than that.
The stuff I've had private healthcare for (endometriosis, varicose veins, plantar fasciitis, back injuries) wouldn't have killed me most likely, "just" made me miserable, immobile, in pain and probably driven me out of the workplace.
Regarding education, the collapse of the SEN system has meant any parent who can possibly afford it is looking towards the private sector rather than tolerate their child not getting a suitable education or no education at all.
Local authorities are paying for some of it because there aren't enough of the right kind of schools and there aren't the staff either.
These are not the private schools one would consider "fancy" (incidentally most of those rely on endowments that pre date the moden education system).
My London Borough has 20% of its SEN kids going out of Borough to be educated, and that's just the ones they know about. They also benefit from a lot of private schools locally meaning most years they are short of 100 or so primary places not the multiple of that they'd be short without the private primaries.
These systems are intertwined. Mess with them in an ideologically based way at your peril!