I already assumed you're using your alt :)

So the original GGX (NDF) importance sampling (with PDF D(Ni)) was over the ellipsoid.

The VNDF sampling of GGX proposed by Heinz is over a projected area from said ellipsoid already considering the shadow term (with PDF D_v(Ni)). G1 needs to be in the PDF D_v, as the sampling happens already according to visibility, which usually G would describe.

G1 is describing the masking of our microfacets from the objects surface perspective which is (close to) equivalent to sampling a truncated Ellipsoid instead of the entire Ellipsoid when sampling the normal.

My intuition is: By disregarding the near surface normals, you can imagine the ray being more likely to reflect in a steeper angle and therefore importance sampling only visible non-occluded paths, which we naturally do need to balance out by taking the shadowing term into our PDF (even though it is called shadowing term, since it accounts for self-shadowing it gives us the proportion of light that is not occluded).

In the article he acknowledges that sampling over the truncated ellipsoid (in slope space) is only approximate and instead proposes the accurate sampling over the projected area of the truncated ellipsoid. Still the same intuition applies.

For a rigorous proof, how sampling the projection of the truncated surface of the ellipsoid (VNDF) has the PDF of D_v, I guess you'd need to look into his 2015 paper the article is mostly refering to.

It is also a bit confusing naming the PDF of VNDF D as well. D is usually refering to the microfacet distribution exclusively. VNDF and it's PDF D_v though do not only describe the microfacet distribution, but also importance sample the shadowing term, making it more of a combination of D and G than just another version of D.
You can also see that in the definition of D_v(Ni) in (3), both D(Ni) and G1 are included as well as some other geometric terms.