The real issue is not really whether Australia belongs to Asia.I think this largely over-rated. When Paul Keating was calling for Australia to be part of Asia I recall the leaders of Indonesia and Malaysia responding to him in a very luke-warm fashion. Infact Mahathir Mohammed openly objected to Australia joining ASEAN.
When Howard took over as leader on a platform of Australia should not lose it's ties to Britain, the Asian countries respected this form of re-affirmation of identification. While I never liked Howard's snivelling and use of race to win votes his decision to confirm Australia's British identity made sense. If anything it increased respect from the Asian region, including China. The Chinese couldn't care if we act like monkeys and scratch our backsides provided we give them what they want.
It has to do with geo-politics.
Each time Asian countries wanted to form a regional grouping of their own, Australia will claim to be an "Asian" country and want to join. This done at the instigation of USA.
Remember years ago Mahathir wanted to form an East Asian Caucus.
He wanted Australia out because Australia is a US proxy.
That was the reason Malaysia was most happy if Australia claim to be non-Asian.
Australia intermittently claim to be part of Asia mainly they want to get involved in Asian affairs as a proxy for USA.
Ordinary Aussies never feel like being part of Asia. They don't understand geo-politics.