Poor communication skills coupled with ignorance has caused many a fight and even wars. The two systems of riding have evolved over the centuries and have developed in order to meet the differing needs of the rider. Neither is system better than the other, they are different.
Stacey Westfall rides with incredible skill wearing a cowboy hat, and without needing reins, bit or saddle. Watch the videos and be amazed.
A Grand Prix dressage rider makes a ride on a horse to music look like ballet.
Stacey would not feel comfortable in a European dressage arena and I don't know of a dressage rider who would ride in an arena without a saddle and a full bridle set.
Spending a day rounding up steers is way beyond my stamina and skill level.
Riding to hounds chasing a pesky fox for a day with the Quorn Hunt in Leicestershire can cost you your spine.
If someone says to me come ride my western trained horse, I'd have to occasionally ask for guidance. If a knowledeable Western rider friend came to the UK and asked to ride my horse, then I'd show him where the pedals and the brakes are hidden.
But, a badly trained, ignorant, heavy handed, coarse, unfeeling, rider shouldn't be let near a good horse, whether it is English schooled or Western trained.