It's really obvious that an assertion is not automatically false just because a person making it used a logical fallacy. If I used a logical fallacy to argue that the earth orbits the sun, that would not mean the earth doesn't orbit the sun. *Duh.*
The Randroids I used to debate with would use the "duh" argument, usually in terms of Kant's transcendentalism. Any time I quoted Kant on their forum, it was either wrong because he was (allegedly) a subjectivist, or it was something very obvious therefore correct by mere assertion. So being obvious was itself basis for criticizing Kant. As a result, almost anything I quoted was either false, or not worthy of consideration because of the "Duh, obviously!" argument. (The truth is that they hated Kant merely because Rand hated Kant.)
And your assertion, "It's really obvious that an assertion is not automatically false just because a person making it used a logical fallacy" is incorrect. A fallacy is either formal or figurative. A formal fallacy invalidates the argument (not 'assertion' as you said, we're talking about arguments here). A formal fallacy renders an argument wrong; a figurative fallacy pretends to use logic when it is in fact a different kind of appeal, one designed to give only the appearance of logic. And I usually consider the argument's conclusion (unless the person is being an irritant). But with fallacies in general, it can be very easy to use them to pull the wool over the eyes of average people.
The way to deal with a formal fallacy is to point it out, and then ask the other person to reconsider their logic and come back later. The way to deal with a figurative fallacy is to say, "While I disagree with your method of argumentation, I still believe your conclusion to be worthy of consideration" (unless, as I said, the other person is being an irritant at which point I will simply use the fallacy to end the discussion).
But if someone's argument is predicated on a fallacy, that does invalidate that particular argument, which means it does nothing to evidence that their assertion is correct. It may be, but the person's argument gives you no additional reason to believe it is. So when you're arguing for a point with the intention of bringing the idea to an audience that does not already believe what you're saying is true, the audience would be justified in disregarding your points if the only ones you have are fallacious.
So if a person does not believe in what you are saying, or already believes what you are saying is wrong, and they dismiss your argument for using a fallacy, they are being perfectly reasonable in doing that. And I honestly don't know if I've seen anyone spell out the idea that they think a statement must *not be true* because someone used a fallacy to argue for it. This seems like smoke and mirrors, a complaint about something that basically never happens, which frankly I suspect is just a pretentious excuse to get away with using fallacious arguments.
I haven't known anybody to be perfectly reasonable about anything, particularly since we are all emotionally attached to our beliefs and will use any method to justify them. The INTJ, for example, will use Fi-tertiary to justify their beliefs on a humanistic basis. It soon becomes evident to a person such as myself that the INTJ is more faith-bound than rational, and that they think in broad terms in order to avoid dealing with particulars. The INFJ, on the other hand, uses Ti-tertiary to bombard me with very specious reasoning which is only convincing to the average person.