I like a versatile horse but have found that there are limits. I have a real nice cremello stallion who is a great Western Pleasure horse, I want to also have him do some Trail classes, Western Riding and maybe some Western Dressage. He's awesome out on the trail too, doesn't just go well in the ring.
He is not ever going to be a hunter type horse ...just doesn't want to go that fast without a LOT of encouragement. So, of course, he's never going to be an over fences kinda guy either.
I think they can be versatile within their scope of what they're good at naturally without necessarily going into every discipline. If they CAN do it, then sure, why not? We need to breed good athletes, more than just to specialize in one thing.