Kids shouldn't be under horses, but I'm not going to say that kids should be around studs because they are studs. There are mares, geldings, spayed mares, and studs all alike that kids shouldn't have anything to do with. My parents operated a breeding farm for years, and the studs were supossed to be as well behaved as the mares and geldings, and if a breeding prospect didn't calm down and work like a "non-stud" they were cut.
A friend still tells the story of coming to our house to look at one of our studs to breed a mare to (he didn't know us yet), and for some reason my brother (< 10 - he's now about 30) was home by himself. He proceeded to show the stud to the potential breeder by climbing up on the side of the stall wall holding the halter out to the stud. The stud came up and put his head in the halter (my brother was too small to halter him from the ground), my brother jumped down and took him out in the barn aisle and set him up like a halter stallion and showed him off. We ended up getting the breeding (and multiple more from this guy) based off the the studs personality and ability. He was also the one that the beginner riders rode, because he was the most trustworthy one we had. He taught my generation in my family to ride, and taught our kids how to ride until he had to be put down after having a stroke.
Our other studs were all nice, but not quite as nice as this one. But hey, they all have different personalities. I know quite a few people that show the horse all year long (including having them in the breeding barn and taking them to a show the next day). But they have it where a certain person hanldes the horse only when it's breeding, and somebody else handles it only when it's being shown, and never do the two cross.
You have to use each horse, no matter the gender, on an individual basis. But any horse can act up, it's just a risk you take (you just try to minimize them)