I wonder if its mostly expressed in the negative form, because "the done thing" by itself is like an incomplete sentence and an ambiguous phrase. Kind of grammatically squishy in some fascinating way.
I don't think it seems incomplete or ambiguous. To me, it is pretty clear that it is turning the verb form of "done" - that is, not the adjective meaning "finished", but the past tense of the verb "do" as in "fishing is an activity that people have done for a long time" - into an adjective modifying "thing". So a "done thing" is just "a thing that is being done". Using that fishing example again, it shifts from "fishing is a thing that is done by people" to just "fishing is a done thing".
This "adjectivizing" technique is a common way that colloquialisms are created. (Maybe it's a bit weirder in this case because an adjective form of the word already exists, but has a different meaning.)
