I sort of agree, but I think you might be simplifying over a couple of steps.

Goth, as we interpret it today, comes most directly from the term gothic literature. Dracula. Frankenstein. Edgar Allen Poe. 18th and 19th century dark romantic literature often with a touch of horror thrown in. The association there was to the gothic architecture - the old cathedrals and buildings - that acted as a characteristic backdrop for those kinds of stories. Think gargoyles and flying buttresses and such. The gothic term there does probably come from Renaissance writers who were referring to the old architectural styles as “barbaric” (or at least very primitive and outdated) compared to the new styles. And, being the Renaissance, was probably a deliberate callback to the Roman era meaning of the word, because they loved their classical revival shtick.