Bold part is completely irrelevant because you can make healthy food both cheap and fast. And it goes against your point about subway, because Subway makes decent food both cheap and fast :P
If you think you can make cheap and fast food that's also healthy, start a fast food chain. You'll be a millionare overnight. Subway is neither particularly fast nor cheap, at least compared to bulk cafeteria fodder.
No, the real problem is that without competition there's no sense of efficiency and quality of service. Compared to what the free market offers, school cafeterias really offer terrible value despite having an advantage of producing in bulk and in controlled waves.
Of course, that's also true of schools as a whole. I went to private school for a while, and it barely was eking by on 2/3rds the tuition per student public schools get from taxes. School services don't have to compete with anything besides vending machines.