Not every stud is a child stomping-woman eating monster! XD
I personally would not allow my little brothers (or, if I had some) children around a stallion no matter how calm. The only horse I let them crawl all over is Sam, and he's older than dirt, and has been strictly tested.
I have met both the 'cute-but-possibly-dangerous' stud and the 'you'd-think-it's-a-gelding' stud. No mean, or evil tempered ones yet. The cute but dangerous was an overall calm stallion. You could pet him and what-not, but when it came to the ladies his mind went out the window. He wouldn't kill you, but he certainly wasn't trust worthy.
The you'd-think-he's-a-gelding is actually the sire of my two MFT. He does pay attention to the ladies and LOVES to show off when theirs a trailer in site, but with proper handling he is not at all dangerous. You could roll all over him and he would just sit there. If he had been more trained as a younger horse I have no doubt a child could play with him without risk of serious injury due to stallion aggression.
Some studs know there job. They know when it's time to breed and when it's not. For example, the studs the kids were washing obviously knew their duty and were trained.
A GOOD stallion is one that can be ridden or led around mares without the risk of an accidental breeding. The number one thing I expect when I meet a stallion is that his mind is not constantly in the breeding shed. I want to see a calm, focused stud whom is safe to handle and to be around. Not one that's near constantly focused on spreading his seed.
Wait! I'll fix it....