All forms of sport or recreation have an element of risk of injury. Just driving in one's car can result in head injury and no helmet worn. I am not saying not to wear a helmet.
Yes you can fall on the noggin and end up like a tomato plant. And maybe the helmet could have prevented the injury.
You could also fall badly and not hurt your head but end up riding in a wheel chair through a spinal injury, so when riding should we wear a stiff back brace, or a cast on the legs and arms to prevent a break. Silly, yes. But now a consideration.
My point and it is not to upset anyone, but if we are so afraid of injury why get out of bed in the morning. Just being alive has its hazards, buying the wife a chainsaw is a potential risk to any red blooded male brave enough to stand up to her who must be obeyed. But I jest.
I don't wear any safety equipment and a cowboy hat. I have at times been a little apprehensive if riding on a gravel road incase I fall, but its not my head I thought about, its the rest of the body hitting the stones.
My opinion for what it is worth, helmets with western style hats attached look a little strange..
If you feel O/K in a helmet then wear it. If O/K in a western hat wear it.
There will be a place for a hard hat when riding, perhaps when working a horse that is unknown etc.
And finally after all of this prattling I have not helped at all have I. Enjoy your dilemma and I appologise if I have stood on any toes.