My understanding is that etymology refers only to the origin of words, while referring to proverbs/phrases and saying we would speak of their origin, not their etymology.
The site etymology tag refers both to words and phrases but the description given from the OED clearly refers only to words.
We also have the tag phrase-origin but it shows no description. Other tags like “proverbs” and “sayings” do not explicitly refer to their origin.
The following old question uses the tag etymology for a phrase, for instance, but there are other instances.
I agree that we may easily assume an extention in the usage of the term etymology that encompasses both word and phrases, but as a site language I think we should adopt a stricter, more precise view.