I think this is a crucial point that is too often overlooked.
Many people think that "free market" means no intervention. While this might be the theoretical definitions, in practice that does not work and the system degenerates.
If a good approximation of a functioning free market can be achieved then it is indeed the best option we have.
Efficient allocations are theoretical (in the sense of not computable within the lifetime of the universe on a real machine) when you move past a handful of actors and a handful of goods. Functional markets don't make efficient allocations - they make allocations that the actors can sustain.
Markets exist no matter what you do. Everything is a market.
Education, for example, certainly is a market. Demand is high and all parents try to get the best the can for their children, which creates an imbalance between demand and supply for the best education. Where school places are allocated according to catchment areas, houses prices immediately reflect the value of certain schools. That's the market talking.
In the UK, there used to be "grammar schools", which were selective secondary state schools. Most of them have been abolished but not all of them. Getting a place means top free education. Result? Insane competition and huge industry of private tuitions and training books that lockout the poorer. Again the market is at play and tells us that a grammar school place is extremely valuable.
Supply and demand, therefore markets, shape everything.