There is NO REASON to be afraid of stallions! What you should be afraid of is their owners!!! LOL! A good owner is fine - the others, fear!
It's not just Europeans who ride stallions. Many show jumpers, dressage and driving horses in the US are stallions. It is all in the attitude of the person training the horse. If the person thinks the stallion should behave like a gentleman and do his job, most likely he will wind up doing so. If the person thinks a stallion is a dangerous, vicious animal, unpredictable and not to be trusted, then it becomes a self-fulfilling prophecy. He keeps the horse in, doesn't train him in different situations, is afraid of a normal animal acting like a normal animal, and there ya go. He's created a monster.
Horses that get a lot of attention and work and activity, are quite happy without being in a pasture 24/7. In fact, 24/7 in a grass pasture is something that's quite unusual in the rest of the world - there just isn't enough land for that. The key is the constant attention and activity they get - and that it's a routine that they're used to.
Some types of show horses in the USA are in their stalls many hours, but not all show horses are.
Most of the better owners find ways to keep them occupied and keep them moving most of the day. There are those big oval exercise lanes and the horses go in those together and seem to relish it(we saw a couple that had learned to hurry to the front of their own compartment, bite the horse in front of them thru the divider, and then wander off looking innocent and guileless, or hurry up and grab a treat from a passing person). There are treadmills, and grooms to hand walk the horses to grass, and swimming, and most of them go out with other horses to trail ride, do hill work, and train in the outdoor arenas. Many of them go out several times a day to work as well. They don't just sit in their stalls. I've seen horses handled like that, and none of them look in the least bit unhappy - not in the least. In fact they look fit, healthy and in great spirits.
Some of the top horses can't be turned loose or they are so fit and sassy they will (or already have!) hurt themselves. When a horse is worth five or ten million dollars, no, a lot of them can't just be set loose or turned out in a group. Even a healed injury or a very slight unnevenness of gait is out. Those competitions include a panel of vets and judges that examine the horse at the start of the competition - even the slightest uneven step, and the horse is out of the competition.