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On itself, a simple claim (like “copyright destroys culture”) cannot be fallacious. It can be only true or false. For a fallacy, you need a reasoning flaw.
Also note that, even if you find a fallacy behind a conclusion, that is not enough grounds to claim that the conclusion is false. A non-fallacious argument with true premises yields a true conclusion, but a fallacious one may yield true or false conclusions.
The issue that you’re noticing with the title is not one of logic, but one of implicature due to the aspect of the verb. “X destroys Y” implies that, every time that X happens, Y gets destroyed; while “X [is] destroying Y” implies that this is only happening now.
In English, the simple present often implies a general truth, regardless of time. While the present continuous strongly implies that the statement is true for the present, and weakly implies that it was false in the past.
From your profile you apparently speak Danish, right? Note that, in Danish, this distinction is mostly handled through adverbs, so I’m not surprised that you can’t tell the difference. Easier shown with an example:
Note how English is suddenly using a different verb form for the second one.