The purpose of the communication in polyamory is to help deal with the complexity of the relationships is my only point.

The total communication may be higher, but I don't think the individual communication is higher than what a monogamous relationship should ideally have. That is, if A and B and C are in a triad then yes there is more communication because there are more relationships, but A and B's level of communication with one another isn't necessarily higher than what it would be if they weren't both also dating C. In practice it often is, but I would argue this is a flaw with how most monogamous relationships are conducted, not a criticism of the difficulty of poly relationships.

Most polyamorists don't want to be married, and most of the people in the US who do want polygamist marriages are some flavor of Mormon or Muslim

I don't think that's true. But we might both have biased samples. I live in the Bay Area and all of the poly people I know are agnostic or pagan. You probably know fewer pagans. I don't think anyone has done a formal survey.

Besides, the polyamorists who do want marriage don't all want the same marriages.

I'm not sure that's true or if so that it's a large barrier. A marriage contract means certain things legally. You can share each other's health insurance, you might owe each other money if you split up, you can see each other in the hospital, you have rights and responsibilities regarding shared children, and you can inherit each other's property, among other things. While poly relationships might differ in their details a great bit, I can't think of why a poly group that wanted to get legally married wouldn't want all the things that legal marriage currently entails or that they couldn't alter anything they disliked, like alimony, with a pre-nup the same way monogamous couples currently do.