Traditional waxed cotton jackets solve the warmth problem by quilting.
Quilted “pockets” in the construction can hold either air or extra padding, depending on how warm the jacket is meant to be. The thinnest, lightest versions, most similar to the Gore-Tex and other “shell” styles we see in outdoor and camping stores, forgo the quilting altogether, using a single layer of thick cotton fabric instead.
Breathability is a natural advantage of cotton. The waxing limits it to some extent, but air still flows much more freely than it can in a synthetic jacket. Waxed cotton jackets tend to be much less stifling than those made from nylon, PVC, or other synthetic materials.
Weight is the primary disadvantage of cotton jackets, particularly if they do soak through and become wet.
There’s little that can be done to reduce the weight, though the best jackets are made from high-quality Egyptian or other long-staple cotton that provides strong, lightweight cloth (think of a very fine dress shirt — it’s the same material, woven thicker).