Basic problem - if something need to be available to those who can't afford what it costs to produce - you either subsidize it or it doesn't exist.
The private sector will not provide a service that can never charge what it costs while remaining affordable to the potential target market.

Your general favorable view of the private sector is BASED on knowing this fact about them. But unfortunately - some things are essential - EVEN IF THEY CAN NEVER BE PROFITABLE.

So society invented a solution for providing those services which otherwise would not exist. It's called taxes and subsidies- and we invented it because, without it, and there is not a shred of hyperbole here: society would have collapsed entirely centuries ago.
Indeed it almost did. Cholera came pretty close to destroying all our great cities. Public sewage systems available to all citizens is the only thing that stopped it.