Afterward, I realized that I have on hand a book that could be described as a very advanced bingo card: Mark Isaak's Counter-Creationism Handbook, which grew out of an FAQ for the Usenet newsgroup talk.origins.
The problem with the Bingo boards is that they're not even a list of "answers to straw arguments" since they're missing the answers. Specifically, feminists treat placing an argument (or even a statement) they don't like on a bingo card as an alternative to answering (or disproving) it. This is similar to the obnoxious debating technique of saying "I don't want to here objection X" without bothering to actually address objection X.
One reason feminists make bingo cards is to say to other feminists, "You're not alone in your frustration at hearing these arguments all the time." Bingo cards function as an expression of support for others in the movement. This seems to me to be a big part of what feminists get out of feminism:
This nicely illustrates the source of the problem: What kind of arguments are the most frustrating? The kind where you don't have a good counterargument (possibly because the argument is in fact valid).
Sure, many people use "I don't want to hear X" or "pfft, X is a well-known fallacy" or "you really should read author X on this subject and come back when you've educated yourself" or many variations on that theme to dismiss arguments they don't actually have counterarguments for. Agreed.
This ought not be surprising... any strategy that knowledgeable people use to conserve effort can also be adopted as a cheap signal by the ignorant. And since ignorant people are in general more common than knowledgeable people, that also mean...
Making fun of things is actually really easy if you try even a little bit. Nearly anything can be made fun of, and in practice nearly anything is made fun of. This is concerning for several reasons.
First, if you are trying to do something, whether or not people are making fun of it is not necessarily a good signal as to whether or not it's actually good. A lot of good things get made fun of. A lot of bad things get made fun of. Thus, whether or not something gets made fun of is not necessarily a good indicator of whether or not it's actually good.[1] Optimally, only bad things would get made fun of, making it easy to determine what is good and bad - but this doesn't appear to be the case.
Second, if you want to make something sound bad, it's really easy. If you don't believe this, just take a politician or organization that you like and search for some criticism of it. It should generally be trivial to find people that are making fun of it for reasons that would sound compelling to a casual observer - even if those reasons aren't actually good. But a casual observer doesn't know that and thus can easily be fooled.[2]
Further, the fact that it's easy to make fun of things makes it so that a clever person can find themselves unnecessarily contemptuous of anything and everything. This sort of premature cynicism tends to be a failure mode I've noticed in many otherwise very intelligent people. Finding faults with things is pretty trivial, but you can quickly go from "it's easy to find faults with everything" to "everything is bad." This tends to be an undesirable mode of thinking - even if true, it's not particularly helpful.
[1] Whether or not something gets made fun of by the right people is a better indicator. That said, if you know who the right people are you usually have access to much more reliable methods.
[2] If you're still not convinced, take a politician or organization that you do like and really truly try to write an argument against that politician or organization. Note that this might actually change your opinion, so be warned.