Existential Comics has a nice series on Fallacy Man, a guy dressed as Zoro who jumps into conversations to point out fallacies. It’s a nice way to show the dialectical error of only pointing out fallacies – namely, that naming a fallacy form isn’t helpful feedback for the argument. You’ve got to explain why a premise is irrelevant, or how some forms of inference are based on incorrect data. Those are all dialectical requirements of reason – exchange. The best part, of course, is that there’s also the problem of the fallacy fallacy. (You’ve got to read to the end of the comic.)
Now, the fallacy fallacy requires additional dialectical baggage, and I don’t see it in the comic posted. Here’s the basic form of fallacy fallacy:
Premise: The opposition’s case for their view (P) is fallacious. (Then the list of the fallacy forms identified).
Conclusion 1: The opposition’s view, P, is false.
Conclusion 2: And, further, my view is true.
Now, so far, just listing all the fallacy forms you identify in the opposition’s case isn’t yet proof that their view is false or that your view is true. BUT: there are a number of considerations that might undercut that. Note, the opposition may have the entirety of the burden of proof. And so, were the opposition to have the view that, say, there’s an elephant in the room, and they can’t prove it except fallaciously, then there’s reason to believe that there’s no elephant in the room. (Otherwise, there’d be evidence). Or consider this in a legal context — all the defense has to do is point out the failures of argument from the prosecution, because the burden of proof is entirely on those who argue for guilty. In those cases, there are default conclusions, and when the case to the contrary fails, we revert to them. So in those cases, fallacy fallacy is no fallacy. To further clarify John’s got a great post on the Fallacy Fallacy Fallacy.