I don't believe it would, either more cheaply or efficiently.
You're arguing against economics there bro... Are you saying that central planning is more efficient in general? Or is there something special about those services which means a free market can't deliver?
I could cite a thousand reasons why - but I'll just mention School meals in the UK.
...provided and subsidised by the Local Education Authority (ultimately by the taxpayer of course)...
Which is the key point. We can debate quality, but these meals were never free. Just like the NHS isn't free today. Somebody is paying, even if it isn't you.
As an aside, I also think competitive tendering is a farce and doesn't constitute a free market. The government buying services on our behalf is no better than them providing it - the consumer must have choice.
What happens to the people who can't afford all the services I mentioned above?
Well you'd think food and clothes would be their first priority, but nobody is asking the government to nationalise these services...
These people would need to rely on charity, as they do now, but that charity would be voluntary rather than coercive.