If it were all as simple as a game of bridge where my fallacy card trumps your argument card, the world would have worked most everything out by now. Thinking in terms of fallacy identification only solves a limited class of problem, in a way similar to that in which arithmetic only solves a limited class of problem. Trying to use arithmetic to solve an algebra or calculus problem will leave you with an unsolvable problem. The answer will be undecideable. If you can get an answer to some simple problem with arithmetic, it will undoubtedly be true, but for most interesting problems in maths you need a much bigger tool kit than arithmetic only can provide.
Translating that idea to fallacy thinking, suppose a climate change denier says “It is simply the height of hubris for arrogant humans to believe they can change the climate.” The argument has a superficial appeal because we admire humility and are taken by surprise by it, but the argument can readily be identified as fallaciously circular. It assumes the correctness of the denier’s position. It is only arrogant and hubristic to believe in Anthropogenic Climate Change if it is not true, a proposition the denier has assumed without demonstration. Other ways of constructing the fallacy inherent in this argument are available. But this approach is only sensible for a limited class of case. And notice that the identification of a fallacy does not tell us whether ACC is true or not. It only tells us that specific argument is of no value.
Suppose I see man A enter a room with man B arguing together. They are alone in the room. I hear a shot. The door opens and A runs off with a gun in his hand. B is dead on the floor of the room with a gunshot wound to the head. Fallacy thinking tends to say things like “The fact that A fled the scene does not mean he is guilty because he might have been frightened”. Or “B might have shot himself and A picked up the gun and ran off in panic”. And so on. The problem is that the misapplication of this style of thinking tends to leave problems like this undecideable. There is no language within it to say “The overwhelming probability, notwithstanding hypothetical conjectures, is that A shot B”. A proposition within the realm of fallacy thinking is either certain beyond argument, or must be wholly rejected, and there is no middle ground.
And the vast majority of real world problems are not of this limited class. Fallacy identification, like deductive reasoning (of which fallacy thinking is a sub-class) applies in limited contexts. Maths. Dumb politics. Unproductive debate about the existence of Og. Remember, identifying a fallacy on the part of a believer does not prove atheists right. It just proves that specific theist argument unreliable. But abductive reasoning is what we rely on mostly. It doesn’t have the promised certainty of syllogisms, but neither does science, which always acknowledges the hypothetical possibility of error. Nevertheless, it is more productive than gotcha arguments about fallacies.
And to address some up thread observations about lawyers (and I am one) not arguing about fallacies in court because jurors wouldn’t understand - that is not the reason. I can’t recall ever seeing fallacy identification as an argument in appellate courts either, although no doubt it has happened. The reason is that the problems involved are more complex than simply purporting to identify a fallacy will solve. If they were so simple that they could be solved that way, they would have been long before the matter even got to trial.