If it works that way, why doesn’t ethanol come off the still entirely and then water? There’s over a 20 degree gap between their boiling points and yet anyone who has ever distilled will tell you that they see a mix that’s at least 20% water at the very start. (You measure as you go along.) This is still well below the azeotropic mix too.

And, later on in distillation, when you’re much closer to the boiling point of water than ethanol, there will still be some ethanol coming out.

I get why you think that, I did too before diving deeper, but I assure you, your mental model of how distillation works is incorrect.

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.