Only because it's at the south pole. If we tilted the earth a bit, it would warm up and start raining. But if we tilted the earth a bit, Australia could probably also get wetter. As far as I understand it, the dryness of Australia, the Sahara etc are caused by the particular location from the equator, where moist air is lifted and brought to the equator and dumped there as rain. So if we pushed Australia further from the equator it would get a bit damper and maybe Indonesia would dry out.

I recommend we don't adopt this program of rotating the earth. It's almost certainly worse than just inventing fission power and pumping desalinated sea water to wherever we want to de-desertify. We'd probably also be better off if we wasted quintillions of dollaeuroyuans on a program to terraform Mars that ends in complete failure and the deaths of millions of workers over a century, but that's also an outcome I recommend we avoid.