The private sector can work in tandem with the Government. If the Government decides it will use taxes to fund heavily discounted child care for example, the free-market can then compete to get that contract. If they are able to provide what is required to a suitable standard - ie, not just chucking kids in a barn and hoping for the best, then they can be paid the relevant money and employ the right level of staff.

There's a half way house between the Government running everything and private enterprise engineering the market so there's a lack of competition and prices stay artificially low. We have some public funded stuff and some private funded stuff over here. We have the NHS as a safety net, but private health cover is also available too.