As a physicist who undertook original research on chaos and dynamical systems at leading international research institutions, I can assure you that your understanding of chaos theory is inadequate, at best.

You certainly can model such systems and people do it all the time. The nature of the questions you ask are different however. Calculating long run averages turns out to be one aspect of such systems for which there are well understood approaches. There is nothing about the chaoticity of the climate which is a conceptual impediment to determining long term trends.