Most of this I disagree with.
I do think there are sound strategic reasons for advocates of same sex marriage not to move on to this; namely that the status of same sex marriage is still going to need to be reinforced over the next decade or so in the US, and remains unrecognized in various moderate to liberal nation-states. That part I certainly agree with. The focus of advocates for same sex marriage need not begin to fracture into other causes in order to achieve these ends however.
Morally and legally, these are not persuasive arguments for the state to be involved in restrictions against polyamorous arrangements. Viewed in a utilitarian sense, I do not see this as a very strong case myself (as a semi-utilitarian, particularly in public policy views). I'm not sure why this is described as a "utilitarian" case against it: that it causes some number of people (who are not party to the arrangements) unpleasantness is not a case for government restriction on its own (it might be a case for coercive restraints of a sub-state nature). A similar and perhaps stronger utilitarian case could be made for any number of existing marriages that they will fail or be singularly unpleasant for one or both parties (or their children, should they have any) for the state to step in and prevent them from occurring in the first place if this is an argument for government intervention. Why this flaw of reasoning does not occur to people in rejecting pluralistic marriage arrangements is very strange to me.
The basic statement boils down to some sort of claim that every person must have equal possible sexual and intimate right to another person. Some people are assholes however. Not every person is easy to live with, or desires to make formal arrangements for their sexual partnerships. Simply because some men or women would be able to amass more partners via their popularity, sex appeal, etc seems like a flawed case for "some men or women would have no access to these partners". This suggests that they would be able to marry or even enter into an intimate relationship with these other men or women in the first place (those who are "closed off" by entering into multiple marriages). Perhaps this suggests some sort of trickle down effect where more eligible candidates for marriage are effectively destroyed because they are bound up in this other arrangement and this eventually screws some number of people. I do not follow that this is likely to be a significant effect.
It also implies that the marriages will be permanent effects. While the divorce rate has been going down (for various reasons), it's still a considerable weapon for marriage arrangements that are deemed unfair or unpleasant to be amended, or for people within those arrangements to revisit these concerns prior to breaking off their marriages, and so on.
Stability arguments at the end are the same gobbledegook that were thrown at homosexual "marriages" being less stable for many years, and in fact to me is and was an argument for recognizing and providing a method of stabilizing influence (a legal contract with social capital behind it, aka, marriage).
As for making extra demands for more people in the marriage arrangement, I'm not sure this is a case against polyamorous arrangements either. People already have the option of engaging in extramarital intimacy and sexuality. When confronted by this, people often find it to be pretty destabilizing to their existing relationships and marriages, often because it occurs without consent or permission or knowledge. This is not an argument against allowing people to try to do this openly or within the confines of marriage. Indeed, I would argue this would be a more healthy social outcome that people who wished to be monogamous could insist on this as a demand (rather than assume it to be the case at their own peril) and people who do not have this as a wish for their private relationships could choose among people who do not make such insistence to conduct their relationships.
The strongest arguments against such arrangements aren't utilitarian but are I think feminist or humanist concerns about consent to such arrangements being often unipolar or overly domination concerned rather than resulting from the active consideration of all parties will and desires for one another. Example: one man deciding on his own to have another wife than his existing partner consenting to this equally as a party to the arrangement, or even that both wives are actively consenting to marry this man in the first place is not clear. This lack of clear consent or equal party to contract is a significant problem ethically (even in a utilitarian sense as it is the source of harm). It is theoretically possible to construct such arrangements in a way that all parties have consented. But it will be difficult to conceive of arrangements for the government to determine this and regulate it (for that matter, the same arguments again apply to regular two-party marriages, as de Boer points out).
The complexity argument for regulating currently standard legal decisions is also a factor. Laws should be simple and easy to transparently enforce in my view. Two-party homosexual marriages are fairly easy to regulate as they're virtually identical in the legal sense to two-party heterosexual marriage in how the various rights and benefits could be conveyed. This is not some invincible argument against polyamory however. It is possible to construct legal rights and methods of conveyance that could account for this. It just requires more work.