The Utility of Polygamy
One reason to oppose polygamy is that spouses have a declining marginal utility, and it's best to set up a roughly egalitarian distribution. Your first spouse adds the most happiness to your life, but each additional spouse isn't going to add much more. Better for them to find other people. It's kind of like progressive income taxation, except that you do it pre-emptively instead of occasionally redistributing existing spouses, since that wouldn't work out very well.
I'm not entirely happy with the idea of analyzing marriage systems by looking at spouses as something to be distributed (as if they were money, or housing, or something), though. In fact, I think some of the problems with traditional polygyny of the FLDS variety is precisely that it treats spouses as something to be distributed as rewards or acknowledgement of prestige etc. But even taking the argument on its own terms, I'm not convinced.
Sinhababu's argument assumes that polygamy necessarily involves an unequal distribution of spouses (or rather, more unequal than monogamy, since a monogamous system involves many single people and many people who have numerous spouses sequentially). If Alan and Betty are married, Chuck and Diane are married, and Evan and Felicia are married, you have everyone with an equal number (1) of spouses. But if Alan, Betty, and Chuck are all married to each other in a triad, and Diane, Evan, and Felicia are all married to each other in a triad*, then everyone still has an equal number of spouses (2). The additional spouse may not double your utility, but it is still increased, and nobody is worse off.
Further, inequality in the distribution of spouses may not be utility-decreasing. Imagine the two triads described above are "V" shapes (e.g. if everyone involved is straight, so there is no marriage between Alan and Chuck or between Diane and Felicia). Here we have inequality, since Betty and Evan have two spouses each while the others have just one. But since everyone has at least one spouse, the total utility is still higher than in the monogamous example.
It is, of course, possible for polygamy to create utility-reducing inequality. The sort of scenario Sinhababu likely had in mind (since it's a common image for critics of polygamy and resembles some of our most familiar examples of polygamy in practice such as FLDS) would be if, say, Alan were married to Betty, Diane, and Felicia, with none of them married to each other, and Chuck and Evan left single. This would indeed have less total utility than the monogamous arrangement (though it would once again become utility-enhancing if Chuck and Evan got married to each other).
But perhaps the second triad mentioned in my initial counter-example isn't working, and so Diane gets dumped and ends up single (or never joined the triad in the first place). How quickly must marginal utility of spouses decrease to make it utility-enhancing to establish a monogamy rule that would force one member of the first triad (we'll say Chuck) to leave (or not have joined in the first place) and marry Diane instead? Chuck's departure from the triad takes away one first-spouse utility and one second-spouse utility from Chuck, as well as a second-spouse utility from each of Alan and Betty. Chuck's marriage to Diane would give them each one first-spouse utility. This means that a second spouse must bring less than a third of the utility of a first spouse for monogamy to be utility-improving in this situation. If, on the other hand, Alan, Betty, and Chuck formed a "V" with Betty as the center point, then monogamy would only take away one first-spouse and one second-spouse utility, meaning that any degree of declining marginal utility would be enough to make monogamy utility-enhancing.
All of the examples so far have assumed that all first spouses bring the same amount of utility to each partner, and similarly for second spouses. Let's consider again the prospect of breaking up the Alan-Betty-Chuck V so that Chuck will be available for Diane. We'll assume that Alan and Betty have vetoed the idea of adding Diane to their marital group, or that Diane would not be interested in such an arrangement (since if she could marry even just one of the members of that V the resulting total utility would be greater than what would be achievable by creating two monogamous couples -- Diane would gain a first spouse and someone else would gain a second or third, enhancing utility along the lines of the example above about inequality being potentially utility-maximizing). Chuck has always had the option to not marry Betty in the first place, or to divorce her for Diane. Presumably he didn't because he derives greater utility from his marriage to Betty than he anticipates getting from a marriage to Diane. As long as his extra utility from being married to Betty rather than Diane is greater than the decline in utility from a first to second spouse that Betty is experiencing, polygamy remains utility-enhancing in this situation. (Alternately, one could claim that on average members of polygamous groups underestimate the utility they would get from breaking away and forming a monogamous relationship with someone else, so they need an enlightened policymaker to guide them to their own best interest as well as the greater social good, but this seems dubious.)
Given all of this, why would Sinhababu assume that polygamy would most commonly be practiced in the "harem" form noted above as utility-reducing? I can't read his mind, but I think the general popularity of the harem prediction comes precisely from the view of spouses as commodities to be distributed, combined with a little dose of belief in differences between male and female sexuality. Thus, we assume that people can be ranked in terms of their desirability as spouses. And we assume that any man who is desirable enough to acquire multiple spouses will do so. If we grant these assumptions we do indeed get a result in which a few desirable men form large harems, leaving a large pool of less-desirable straight men without wives. (The usual model is very heteronormative, so it's hard to say exactly what its premises entail for non-straight and non-binary-gender people.) The harem model is so common in real-world examples of polygamy precisely because societies like modern FLDS or ancient Hebrews did treat spouses as commodities in this fashion and did make strong distinctions between male and female sexuality. If we remove the assumption that only men are interested in harems, then all of our unpartnered straight men could go join a desirable woman's harem, eliminating the utility-reducing pool of unpartnerable singles. And if a culture does not think of spouses in commodity terms, then 1) fewer people will choose polygamy, and 2) those that do will tend to follow the more egalitarian models used in my counterexamples.
*My use of triads and Vs (which could be extended mutatis mutandis to quads, quintets, etc) is based on a very traditional definition of marriage as a commitment that (among other things) forms a household. Household-forming marriage implies that marriages, no matter how many of them an individual has, will lead to the creation of a series of separate household groups. If, however, we allow a broader definition of marriage that entails only a commitment to mutual support, we could end up with a wide-ranging network of spouses that in theory could eventually link in every polygamous person in the society.