I think the essential factor would be stability of a core set of adults taking care of the children. Those poly tangles that have long-term stability among most of the adults would likely be best for the children. Those with a high turnover rate among partners who aren't biological parents could still be good families for children (if the parents are strong and capable), just not as good as having long term engagement by most of the adults who have been involved.
As for families operating under other forms of non-monogamy, I think those are situations much more like mono families. Most open couples aren't going to expect any other partners to act as parents. I suspect parents who swing are more likely to hide their activities from the children.
So, a stable core of strong capable parents would be the key element in providing a good home for children--with only one being necessary and value added with each after that. I don't know if the typical woman knows enough about poly to even begin to make that connection, though.
The same phenomenon applies in terms of social support for the woman, too. Women do better the better their support systems, so a larger support system from a poly tangle would be of benefit. Again, gaining enough knowledge about poly as a viable choice would be required prior to that connection getting made.
Folks seem to be adept at creating fairy tales that work for their own situations, so I doubt replacing all of the traditional tales would have to take place. I suspect it will just take broader knowledge of how poly relationships can work out for women to consider whether or not they could do poly tangles/families.
When speaking of various forms of non-monogamy...it ain't poly if you're just fucking around.
While polyamory, open relationships, and swinging are all distinctly different approaches to non-monogamy, they are not mutually exlusive. Folks can, and some do, engage in more than one of them at a time--and it's all good.