You make some good points running dog. I do like the idea of dropping the treat for a really rambunctious dog.
I also agree allot will depend upon the behavior of the individual dog in making a determination of one's personal comfort level and preference.
Interestingly Veronica stopped mouthing as she got older and only retained the soft mouth aspects of the training...I'm guessing she just grew out of mouthing in general. Also, she never mouthed anyone outside of the immediate family...and now I can't remember why lol! (In my defense it was 8 years ago). She was socialized ALLOT...but I think she only did the mouthing when she was over-tired and fusing so that was always at home with us, or when we were putting on her collar, leash, coat...which we were the only one's doing that.
I believe she mouthed in play also...and again, all interactions with people outside of me and my husband were highly supervised, not because I didn't trust her, but because I didn't trust the reactions/perceptions of people who weren't us (I was just a super over observant, micro manager because of the "pit bull" designation). Plus I may be in the minority here but I never thought dog/puppies of any breed and very small children were a great combination for just the reason you mentioned, as well as accidents occurring because there are people who think their children should be able to do whatever they like to the dog and that the dog should have an infinite amount of tolerance. I am in the minority I know, but I disagree. I think if you touch the stove you get burned, if you pull the dog's tail, you get bit. - good life lessons all around - there should be a responsible adult there making sure lessons are learned without any of the negative consequences actually happening. That's called parenting. (Sorry for my rant, you can tell that's a subject I can get on a soap box about
But I do agree with you about the perceptions of modern society. Veronica is extremely out-going, loves people and loves to interact with them. But she only does so with my permission, while on leash while I'm standing there monitoring the entire interaction (unless I know the people VERY well, I know their approach to dogs and I know their perceptions about dog behavior and they are adults - then if we're in accord on all those things - I trust them to be responsible and interact with Veronica.)
In other situations, walking on the street, meeting strangers I have trained her to ignore baby carriages and small children and focus on me. I don't want some hysterical person yelling "Oooh that pit bull smelled my baby" - plus I think it would be rude, not everyone likes dogs and I totally get people not wanting strange dogs around their children. I only allow Veronica to interact with children who's parents ask if she can. If the child asks I make it a point to say to the child "only if your mom/dad says it's OK" while looking at the parent. (Of course there are those occasional sad cases where the child has already thrown themselves around Veronica and the parent is no where in sight...). When given permission Veronica will give some licks and kisses and then I thank the parent/child for giving her attention and we walk away.
But I think in her case, allowing the mouthing helped her to develop a soft mouth but it didn't encourage her to maintain the mouthing beyond the stage where it would naturally fall off anyway and she was developing more "grown up" behaviors.