I mean, that can be said for literally everything, no? What kind of proof would you need? Everything about languages and the evolution of words is studied by experts, and I am not one of them, so what else can I do but take their word for it?

If you look up their ethymology online, they are both from the same word. The wikitionary entry also claims it does come from it. https://fr.m.wiktionary.org/wiki/gargouille