You generally treat mild inflammation by cooling. The reason why you don't want to use ice water is because that will cool the outer layer of your skin to the point of pain before it's cooled the deeper layers significantly. Using slightly cold water means you can stay in the water for fifteen minutes easily, cooling all the skin layers well.

I thought this was common knowledge, i.e. basic first aid for burns in general?

My point was merely that "basic physics" doesn't tell us much here. You have to know that it's really inflammation that you're treating. You have to assume that cooling is the only contribution of the aloe (clearly, cooling isn't the only thing that can treat inflammation).

I mean, I know what's recommended as basic first aid for burns, and maybe that's common knowledge. The why of it, in enough detail, almost certainly isn't. Note also that aloe is quite commonly recommended as a part of "basic first aid for burns in general" following cooling.