They can be necessary but not sufficient conditions for buying a hat. The issue is that even once those conditions are met, those two reasons aren't merely reasons for buying a hat, but important specifications for the hat you are going to buy. If your goal is to prevent heat stroke, a hat that keeps you warm in the winter is counterproductive.

(I have owned quite a few hats, and used to wear one most of the time in public - in my college years a bowler, later a fedora, by which I mean a fedora and not a trilby. I donated most of them after realizing the effect they were having on my mood. At this point I own two hats; one is a black oilcloth hat for rain and costumes, and the other one is a fabric hat for sun.)