The irony of using an argument about stability though, is that it assumes that monogamous marriages are somehow more stable than poly ones would be. While divorce rates aren't as high as widely believed those who do divorce often remarry, which obviously requires dating again. Sometimes adults get divorced and remarried multiple times. Children can have many 'parents' over the course of their lifetime. So why is that somehow more stable than a polygamous marriage with multiple partners who may come and go as well?
Formalizing a relationship doesn't change any of those already existing qualities. Rather one could argue that marriage can offer many protections to women from such circumstances. Rights, security, stability, but also power to end the relationship with a no-fault divorce. Divorce and enforcing rights is often not simple and can be arduous. If your issue with polygamy is that women aren't protected enough or that marriage doesn't protect women enough, then advocate for expanding protections because thats what you really take issue with. Not polygamy.
On the surface this might be enough to give people pause. Because to many, logistics acts as a reality check. Non-monogamous people, no matter how much of a right they might have to marriage, simply don't fit and that's that. But why? If we can accept that there is a right to marriage how can logistics really be enough to stop people from fulfilling this right?
We've seen these kinds of arguments before and they clearly didn't stand up to the test of time and reason. With other rights we have dealt with serious logistical issues and come out the better for it. From liberating slaves all the way through to fully ensuring their right to vote (which we are still working on) and even all the efforts that went into ensuring women had a fully respected right to vote we have made significant logistical changes as has been necessary to our system. These required changes in law, and tax codes, and inheritance and divorce and many more things. Recognizing rights is often difficult and requires that we as society and our governments put in effort, monumental at times, to ensure their protection - it's an ongoing process and one we should never stop. That's what makes rights so special and dear.
Are there serious questions that need to be answered? Sure. How exactly should we structure taxes? How would divorce work? What about custody of children? Should there be limits on the number of people? All important questions. Logistics plays a role but it plays a role in deciding how we enact things not in deciding not too. Will it be complicated? Probably. Messy? Maybe. But everyone who has multiple partners should have the option to marry their partners just like monogamous people do.
The bottom line comes down to this: We as a society value people making free choices of their own accord and don't want anyone interfering with that ability. We make exceptions when the decision goes against a specific state interest (and this must be a particularly strong interest) or if it is a harmful decision. Does the state have an interest in preventing polygamy? Not specifically and certainly not under an ancillary interest such as protecting minors or women etc. What about causing harm? Who does it harm to allow people who freely love multiple people to choose to marry? To use an adage from the same-sex marriage debate: if you don't like polygamy then only marry one person. Nobody is forcing anything on anyone. It would simply be available for those who want it. So really - Why not polygamy?