The US gives us the perfect sandpit to test this theory.
Their healthcare provided by corporations is vastly more expensive and has much worse outcomes than healthcare provided to billions of people by governments.
But US healthcare is also vastly more expensive and has worse outcomes than healthcare provided privately in much of the developed world; most of the developed world doesn't use sole government provision, and much doesn't use sole government funding; lots of places have private providers and private insurance with a government backstop (conceptually like the US system, but without the holes.)
Their healthcare provided by corporations is vastly more expensive and has much worse outcomes than healthcare provided to billions of people by governments.
Same for higher education.