Logical fallacies don’t necessarily disagree with facts. While the most common examples are simply unsupported statements that sound supported, very often we don’t have the luxury of working with clearly factual statements as a basis.
All rhetoric is at the end of the day a fallacy, as the truth of the matter is independent on how it is argued. Yet we don’t consider all rhetoric invalid, because we can’t just chain factual statements in real debates. Leaps of logic are universally accepted, common knowledge is shared without any proof, and reasonable assumptions made left and right.
In fact one persons valid rhetoric is another persons fallacy. If the common knowledge was infact not shared, or an assumption not accepted, the leap in logic is a fallacy.
I would try to focus less on lists of fallacies or cognitive biases and more on natural logic. Learn how to make idealised proofs, and through that learn to identify what is constantly assumed in everyday discussions. The fallacies itself don’t matter, what matters is spotting leaps in logic and why it feels like a leap in logic to you.
After all, very often authoritive figures do tell the truth, and both sides of the debate agree on general values without stating them. If someone starts questioning NASA or declares they actually want more people to live in poverty, they did infact spot very real logical fallacies in the debate, but at the same time those fallacies only exist from their point of view, and others might not care to argue without such unstated common ground.