Sugar does not kill heat, or it would make no sense to put sugars in hot sauce.
The way it "removes" heat is you eat raw granulated sugar and rub your tongue against the roof of your mouth to scrub the oils off, as sugar has surfactant properties, but you have to scrub, like washing clothes by hand or dirt off your hands (surfactant detergents). Once it's dissolved into a liquid you just eat it's not going to do anything at all. Like pouring Tide on your jeans and waiting.
Milk only barely works, nothing noticeable for extreme heat. It's the casein in milk.
What really works is time. Let that fire burn out!
I say beer works, it's a lie, but when you're really drunk, well, it kind of does.