It's important to know where errors in reasoning are made. Fallacies can indicate such errors. But sometimes people can arrive at true conclusions despite using minor errors in reasoning to prove their point.
Does the use of a fallacy automatically mean the conclusion or proposition is false? No. Thinking that a fallacy invalidates a proposition is an error in reasoning known as the fallacy fallacy.
The end. That's what the fallacy fallacy is. If you want to understand why this is the case, please continue reading.
What's more important: getting to the right answer, or being "100% perfect" about how you arrived at such a conclusion? Some people just ignore the conclusion itself and want to point out fallacies that were engaged in yet don't affect the veracity of the conclusion.
People can become fallacy-obsessed, and think a fallacy demonstrates a falsity in someone's conclusions about reality. But it actually only demonstrates a falsity/error in reasoning -- not an error of the conclusion itself, but in explaining the conclusion.
The fallacy in explaining a wrong/erroneous conclusion can indicate that the fallacy is how someone made the "wrong turn" and arrived at the wrong conclusion. Recognizing fallacies is helpful to show people why their conclusions are false -- if they are indeed false -- because their reasoning is faulty as indicated by a fallacy.
But someone can use fallacious statements that are erroneous in some way, while advocating for a verifiable truth. The conclusion they are talking about is true -- or at least contains a kernel of truth -- but the way they delivered the concluding truth had errors. Despite some minor errors, most people can still understand the truth that is being conveyed. People can also accept partial truths mixed with falsity as a whole, and take everything to be "true" when it isn't. Fallacies can help prevent that, but verifying the conclusion is also required in order to not be fooled by convincing words.
Fallacies are useful to show errors for why someone has accepted or arrived at a false conclusion, but it's not the be-all-end-all to validate a conclusion. You can even structure something logically without any fallacies yet purposefully propose a false conclusion -- or a conclusion that makes no sense. The logical construction of an argument doesn't automatically provide truth about a matter.
The lack of fallacies doesn't automatically validate a conclusion, nor does the use of fallacies automatically invalidate a proposition or conclusion.
A person trying to understand someone else can get stuck in the message delivery -- the rhetoric used -- rather than looking at what is said. Being able to explain the answer properly is important, but not as important as arriving at the right answer.
If the conclusion is erroneous, then fallacies can be exposed to show how the person is not reasoning accurately to arrive at the conclusion. But to simply claim a conclusion is false just because there are fallacies, is a fallacy fallacy.
"This is not true because you used an 'ad hominem'."
"This is not true because your conclusion still 'begs the question'."
"This is not true because of X, Y, Z, fallacy."
Using a fallacy doesn't make something true, just like using a fallacy doesn't make something false either. Fallacies are indicators that the conclusion might be false because fallacies were used, but not that it is certainly false.
To claim a conclusion is false by looking only for fallacies in message rhetoric, without even looking into the conclusion honestly -- without curiosity, critical thinking and objective detachment from ourselves -- is a fallacy-error of the mind itself. This is the fallacy-fallacy.
I like to call the rigid myopic use of logical methods anal-logic, a false logical process. The truth doesn't really matter to these types of thinkers. Anal-logic is not real honest logic. Look at the truth that's being expressed if it's there, not the rhetorical form of the message alone. Logic can be used to do wrong things, like being "cold hearted" and harming others, but also to argue dishonestly by not looking for the truth.
If someone expresses a fallacy but speaks the truth about reality, then it's still truth despite the road taken having issues and not being the best path to get to that conclusion or reality.
A rhetorical focus on the message delivery first, and not truthfulness of the conclusion -- or only on the message while disregarding the truth in a conclusion -- is indicative of the poisoned version of the Classical Trivium. Classical rhetoric favors effective convincing of a point, true or false. Winning through effective communicate to convince someone. This is when the focus is on rhetorical manipulation through cunning word magic and sophistry, rather than arriving at the truth. Someone's message delivery can lack being the most effective, while still conveying the truth of the matter.
When you only point out the fallacies, while ignoring the truth or falsity of the conclusion, you tend to side-track the actual issue. Fallacies need to really be pointed out when they prevent access to truth. This way the false paths used to arrive at the false conclusion can be corrected by removing those errors in reasoning.
Messages used to express a valid truth can be erroneous in their expression when they contain fallacies, but the truth within is still at its core -- inside, internally, in the substance of the message.
So yes, sometimes people fail to express things accurately, and you can point it out. "This fallacy", "that fallacy", etc. If they are expressing a truth, but do it incorrectly -- as with a fallacy -- you can't always simply deny their conclusion while only focusing on the fallacy used to demonstrate it.
Clever manipulators (as well as some people in error) like to wear others down with endless questions, doubts or demands. They can act like they want the truth of the matter, or that they want to be corrected, yet trying to correct them or answer their questions is like banging your head against a wall. They often get you to go around in circles, never ceding a point to finally arrive at the destination (truth) of understanding something more. I've had many of these arguments so I can often detect people who use similar tactics and avoid getting involved in using up my time and energy.
They don't want to understand or be corrected, they just want to play manipulative false anal-logic games and drag you along to waste your time and energy "debating" them. This is the dark logic of false rhetoricians who engage in clever sophistry to "win". They claim to want reason, but they don't really. They just want to hold onto what they were holding onto -- be it a falsity or denial of a truth.
When someone doesn't really want to arrive at the destination of truth by engaging in pseudo-rational debates, is it really worth your time and energy? I think not. Choose where you put your time and energy. Time is limited in life. You don't have to convince everyone, let alone waste time on those who don't really want to understand.
TL;DR?
If someone's conclusions are false, detecting fallacies can help show why someone arrived at a false conclusion. But detecting fallacies doesn't mean someone's conclusions are false: that's the fallacy fallacy.