I think the conservative groups are right that having society OK homosexuality is likely to lead to society's eventually OKing polygamy. At the moment I'm not convinced that OKing polygamy would necessarily lead to problems, although on the other hand I can see how it could lead to much-more serious problems than the forms of inequality the media normally pays attention to (eg inequality of status via fancier houses / cars / clothing).
I think you may be able to draw a good analogy with money: imagine you're in a society where everyone can only make a certain amount of money per year (say, $50,000). Everyone makes the same wage/salary, although you have the option to quit your job and not make anything. Would it cause society to fall apart if you made it OK for some people to earn more money than others? Most societies in the world have a system-of-inequality like that and it seems to be working out fairly well. I think the important thing is to make sure that everyone is able to get access to enough money to meet their basic needs, and then if people want to trade more of their time for more money (eg giving up Netflix to spend more time at the office), they should be able to do that. So, for example, if it were to get to a point where rich men had so many wives that poorer men couldn't get a wife, you might start to see social upheaval.
In addition, keep in mind that even so-called conservative-values guys like Rush Limbaugh and Newt Gingrich have gone through multiple wives; it seems to me that's basically polygamy.
2010.10.19 - WIRED - How Beer, Oprah and Sergey Brin Can Help Cure Aging
Aubrey de Grey: It’s public knowledge I am polyamorous as well. That’s something that goes down not so well with some of my more politically sensitive friends and colleagues. But it goes quite well with some other people. [laughs]
(...)
Aubrey de Grey: We’ve got obviously a much greater incidence of serial careers and relationships, just because people are living longer and having more opportunity to do different things in their lives. That definitely seems to be something that’s going to naturally extend. And in relationships, well, I just told you I’m pretty adventurous that way. That’s actually something I feel quite strongly about. The whole monogamy jealousy thing is a bit archaic, really. [laughs] It probably won’t last very long.
2015.06.26 - Politico - It’s Time to Legalize Polygamy
2015.10.26 - NYT - Not Enough Women in China? Let Men Share a Wife, an Economist Suggests
One solution is to have several take a wife together. That’s not just my weird idea. In some remote, poor places, brothers already marry the same woman, and they have a full and happy life.”
Polyandry has been practiced before in China, particularly in impoverished areas, as a way to pool resources and avoid the breakup of property.