In general, a Gaussian is no longer a true Gaussian after camera projection since the pinhole camera projection function is nonlinear (due to dividing by z). However, if the Gaussian is small relative to the size of the image, you can apporximate it by linearizing the projection function. Therefore the Gaussian splatting paper uses the Jacobian of the projection function as described in equation 5 of the paper [0]. In practice, this approximation is extremely good. This Jacobian is the matrix you mentioned in the third link and it is mathematically sound and not "manually counter balanced". For a derivation, see [1].
I read the paper and I am aware that the gaussian projection is an approximation anyway (hence I spoke about ellipsoids, not gaussians). Still, one could at least aim to get the iso contour right and yes using the Jacobian matrix is not unsound, just incomplete. As I said, this approach can not produce the distinctive "wiggle" that you get from rotating an ellipsoid while staring dead center at it.
[0] https://repo-sam.inria.fr/fungraph/3d-gaussian-splatting/3d_...
[1] https://math.stackexchange.com/a/4716514/43771