I think I'm starting to see where both "schools" see eye-to-eye...
It's just like that whole "poly relationships are just like mono relationships, but with more people" thing.
The concerns the poly parents here are describing echo what any monogamous "good parent" would say when looking for partners. They don't tend to get together with the other parent and decide a-priori what future relationships will be like, when they'll date or whether they're "allowed" to fall in love. But when someone does come along, if one parent has a concern about whether that new boyfriend is a good influence on their child or if they're distracting the parent so much that the child starts to feel neglected, they tend to bring it up.
To that end, my step-daughter's monogamous mom recently "dumped" (daughter's words, not ours) her at our house for the weekend so that her new boyfriend could come spend some time without the kids around. But she brought her to our city, 1 1/2 hours away from where she lives, for a "shopping" trip and then told her when she was here that she had to go to her dad's for the night, no toothbrush or pajamas... This is meant to illustrate that none of the issues brought up by parents here are unique to polyamory, they apply any time any parent is dating anyone, and monos can fuck it up just as bad as we can. At least if one parent of a poly relationship gets a little bit lovestruck and distracted for a while, there's another parent living there full-time who's on top of it and looking out for the kid. Meanwhile, my step-daughter is feeling completely neglected and abandoned, and there's not much we can do because she doesn't want to live with us. And her mom never listens to us when we voice our concerns, she tells us to mind our own business (as if the daughter wasn't "our business".) But if she were our partner, she would "have to" listen to us :P
So as far as kids are concerned, and "making rules" to prescribe how new relationships will look: I can see how that would be a problem and bother new potential partners. And as was said previously, if a new partner is a genuine "cause for concern" in that they may try to come between the existing partners, or may be negative influences on the kids, and if the partner dating them is too caught up in the moment to see that, then all the rules in the world won't mean bubkiss. But if the relationship is healthy and there are good lines of communication, then formal rules are unnessesary.
For example, I somewhat doubt that redpepper ever sat down with mono and explicitly said "look, I have a kid, and if he ever needs me, I may have to cancel our date." Maybe she did, but mono seems like a very reasonable and responsible person, and I just don't think it was necessary to say that. And if she did say it just to make sure, then I'm sure he said "well obviously!"
“As I am sure any cat owner will be able to tell you,
someone else putting you in a box is entirely different
from getting into a box yourself.” —bisexualbaker