Did Aboriginal Australians use fire in their hunting practices? Yes. In some areas of Australia, they still do. However, I take issue with your friend's description of the practice as wanton or that such practices have contributed in any significant way the desertification of Australia. While not in regards to desertification, indigenous use of fire in landscape management here in North America has similarly been called wantonly destructive, much to my annoyance.

Fires happen naturally and naturally fires tend to be larger and more devastating than small hunting fires employed by humans in various parts of the world. Hunting fires tend to be used seasonally, when people have greater control over them and there's less risk of the fire going wild. Such fires create a clustered patchwork of successional species, increasing local biodiversity and providing animals with a mix of foraging and sheltering areas.

This lecture briefly discusses the practice of hunting fires among contemporary Mardu in the Western Desert. For a longer treatment on the topic, I recommend The Biggest Estate On Earth: How Aborigines Made Australia.

As for why Australia is really as dry as it is, it's largely a matter of geography. It's largely located within the southern Horse Latitudes, which are typically hot and dry. Most of the world's deserts are located within these latitudes. For comparison, most of Australia is within the same latitudes as the Sahara (just a different hemisphere).