"If I make a batch of lemonade for £100 and sell it for £200 I have made £100 profit (ignoring any other costs to keep it simple). If I manage to make the next batch for £90 and sell it for £200 I have made £110 profit - I have increased my profit by reducing my costs. Is that really so hard to understand?"
No, that is easy to understand.
But in your example you are making something and selling it.
That's where your profit comes from.
If you can cut costs, you can make more profit.
But that is nothing like what "profit making" schools will do, unless they are charging fees.
The plan is more akin to a situation where every child is entitled to free lemons and you take ownership of previously publicly owned lemon trees and get given public money for handing out the lemons.
The lemon distributors are all trained using public money.
Children must be given free lemons, but they are also required by law to eat a lemon every day until they are 18.
So if you spend less than you are given of the public money you are given for providing the lemon service you are saving money.
But you are not creating anything even close to a profit.
Except in neo-liberal land where we all pay our taxes so public money can be given to corporations.