I'd wait for the current struggles over gay marriage to die down
It has died some time ago in more socially progressive countries, but none recognize a group marriage, so, I agree, clearly social mores are not the only or even the main issue.
My guess would be that, just like in Physics, in Law the general n-body problem (ahem) is much less tractable than the two body problem. I suspect that there are many relevant legal issues that have not been even touched and will have to be addressed, before a reasonable group marriage framework can be constructed. Child custody/support/rights, potential for spousal abuse are some of those.
On general grounds, one would also expect the stability of such a union be much less than that of a 2-person one, thus reducing the need for legal protection. Most polys freely admit that they have a "primary" and one or more secondaries, who are not in an equal position. Again, the analogy with orbital motion is quite interesting, but I will not push it further.
I don't have statistics on poly relationships, and I doubt that anyone has good statistics.
Still, so far as raising children is concerned, I think you're conflating two situations. A poly relationship might be considered unstable if one or more people leaves, but if a core group remains raising children, it might be more stable for children than a couple that breaks up.
A thought occurred to me today as I skimmed an article in a rationality forum where the subject of gay marriage cropped up; seeing as the issue has been hotly contested in various public fora and especially the courts, what about poly? After all, many if not all the arguments for gay marriage apply to poly marriage as well.
Questions for LWers who are currently in a such a relationship, or have an opinion to share:
Do polies want to marry each other or do such relationships not lend themselves to permanence above a threshold of partners? Should polies campaign for the right for a civil union anyway? what are the up and down sides of this? etc