The phrase to make the most use of every minute isn't idiomatic; it's wrong. Nobody would say that.I know it is right to say: make (good / better/ full) use of every minute or make the most of every minute. But is it also idiomatic to say :make the most use of every minute?
You seem to be suggesting that the first phrase is idiomatic as well. I wouldn't say so, although it is correct.