Flight of the Conchords could not be less Australian, in terms of its vibe and sense of humour.
My newly found favourite guy here in QLD (father of school friend of my twins) is incredibly self-deprecating. At English/Irish, or NZ levels. But you are right it's rare in Australia. The brutal un PC put down, the more common humour take.
Australia's version of comedy news show "Have You Been Paying Attention" (the original) is a no holds barred, not much off limits effort. The show's host (Tom Gleisner - Working Dog Productions) goes back a long way in Australian comedy. For sure if it was in NZ, alot of it would be censored. Melanie Bracewell, Guy Montomery, Ray O'Leary, Urzila Carlson & Hayley Sproull are often on the Aussie version. Ray an easy target. I'm sure they notice the difference.
The much loved, brilliant John Clarke (RIP & considered by many the best comic ever based in Aussie) made Kiwi comics popular in Oztraylia. Maybe Palmy North's best ever export. Sorry off topic