One of the problems of children at weddings is that parents often forget to parent, either because they've been drinking or because they're having a good time catching up with friends & family. They also might assume that someone else is keeping an eye on the kids.
I think one of the biggest potential problem is that people have different parenting styles which can lead to friction, which no-one wants at a wedding. But if anyone intervenes it's WWIII.
DB's wedding, his friend's kids were throwing metal cutlery at each other across the table during the speeches, I took a fork off the kid sitting next to me before he could throw it at his brother and their Dad just shrugged and said "It keeps them quiet".
Friend's Summer wedding, posh hotel - kids climbing on statues, running through flower beds, pulling on draperies etc. Parents oblivious.
At my own wedding, I had to intervene as my SIL had threatened to punch my 70 year old auntie's "lights out" for stopping DH nephew and niece playing tag around the band's equipment (band members had already asked them to stop to no effect).
If I was getting married today, I just couldn't do with the hassle