Further, I’m pretty sure the result is not “X has to cause Y” but “this distribution has measure 0 WRT lebesgue in models where X does not cause Y”

Yes that’s true. Going from “The distributions in which X does not cause Y have measure zero” to “X causes Y” is I think common and seems intuitively valid to me. For example the soundness and completeness of d-separation also only holds but for a set of distributions of measure zero.

Fixed, thanks!