Not just wrong, they are totally different stars. Good luck finding the Big Dipper down under!
Maybe because I've seen northern hemisphere sky for 40+ years, the southern hemisphere sky is much more interesting to me. Just seeing Large/Small Magellanic clouds and realizing those are the closest galaxies to the Milky Way. Seeing Alpha/Beta Centauri and knowing those are the closest stars to our Sol. Just kind of different cool
On the other hand, having a Pole Star is a great idea. It’s not _that_ hard to construct south out of the Southern Cross, but a star (close enough to) on the pole is smart…