I was never required to wear a helmet by my parents, but 4-H required me to do so. Let me tell you that it saved my head many a times! I still wear one even though nobody is telling me I have to. I feel naked without it. My dad wears his cowboy hat when he rides and I tried that once, but I hated it! I just felt so unsafe. You just never know what a horse will do, and on a trail those risks are even greater!
Back when I used to show, competitors wore cowboy hats in mounted western classes in the 4-H shows. I was actually happy to see them change the rules so that you HAD to wear a helmet during any riding class, not just speed events.