This is more about fact it is other people's father etc and I don't think it is appropriate for their grief to have the additional complication of toddlers running around.
This. My father's funeral was ruined by a reprobate of a distant cousin who thought it would be fine to bring her small child and then allow him to screech throughout the entire service without even attempting to quiet him or take him out. My final memories of my father are not of a lovely peaceful time bidding him farewell, they're of fuming at my piece of trash relative who clearly gave no hoots about anyone but herself.
I do not think young children belong at funerals. By their very nature, children demand (and deserve) attention. At a funeral, the attention is and must be on the deceased. To bring a child who then has to be seen to is just drawing the focus away from the point of the day, and that, to me, is so incredibly selfish I can't even comprehend it.
There are those who will say "Well, little Johnny had a relationship with Great Uncle Mervyn too" well that may be, but Little Johnny is three, doesn't understand why everyone is so sad, isn't going to understand no matter how much you explain it (he is too young to comprehend death and its enormity), and honestly will gain nothing from being at the funeral. All Little Johnny understands about the day is that he's being told to sit still and be quiet for a very long time, which he didn't sign on for and doesn't want to do. All the other funeral goers will remember is "Hey, remember at Great Uncle Mervyn's funeral when we couldn't even hear the eulogy because Little Johnny was creating and Cousin Tessa wouldn't take him out?"