As a newbie to both western and showing this year, I've competed more in a hard hat than in a stetson - and wouldn't dream of not wearing a hard hat for warming up/schooling or hacking out. And no instructor has ever allowed me to get on for a lesson, even with my own horse, without a hard hat on.
At shows I've not once been made to feel either stupid or novice for not wearing a stetson; my decision on which hat to wear has been governed by how confident I am on each occasion, on how settled my horse is and - not least this year(!) by what the weather's doing!! And whether I've worn a stetson or hard hat has made not a jot of difference to results - when I've done better it's because I've ridden better, not because I've looked any better!
That said, I love the fact that here is an equestrian discipline where you can really express yourself through your attire and your approach to grooming, and the care that western riders take with their gear and their horses does put most English competitors to shame. It really does look fabulous (or am I biased now??!

)