For interactive navigation on a sphere (remember, this is just a sphere looked at from the inside) you actually want absolute euler angle based rotations, otherwise compound rotations around any closed loop on the sphere's surface will change the orientation of the camera and make further navigation extremely confusing. Source: many years spent designing 3d applications. Also, Google Earth and every other major app do it the same way.

Im fact, if you zoom out all the way, you'll see that we're actually looking at a sphere from the outside, as if the stars are situated on a celestial globe. It seems to be a relatively common convention for interactive night-sky maps.

Inverting the Z axis purely as a UI quirk/implementation detail is one thing, but more interesting would be to embrace it as a means of representing the time scale of the universe - as you zoom into the earth, you zoom into the deeply redshifted ancient data, which conveniently is from a smaller universe so projecting it onto a smaller shell of the sphere makes sense. Until you reach the core, which is a picture of the cosmic background radiation.