You're flipping the ingredients here. The vapor will contain some amount of the liquid that is NOT at boiling point. When you distill ethanol, the vapor contains some water. When you distill methanol, the vapor contains some ethanol. The output contains both.

That says nothing about the remains of the input. When removing methanol we are trying to get it out of the input side. So while the output will indeed contain both, that doesn't mean the input will still have any methanol.