I think you're mistaking "giving them carte blanche" for "highlighting one mistake they make in a long post containing a lot more information, context and nuanc

I think you're mistaking "giving them carte blanche" for "highlighting one mistake they make in a long post containing a lot more information, context and nuance without acknowledging any of that in the slightest, and use that as if you caught them red-handed in a gotcha! that breaks down their entire argument". I guess that's a specific version of the "cherry-picking data" fallacy.