My own experience has been that I see a GP who sometimes bills me and sometimes does not, depending on reason for visit - was bulk billed for prescriptions / receiving results / very short appointments, but paid a gap for investigation of a dodgy knee. The X ray I had to pay part of but then the private consultant and public MRI were bulk billed.
When I was unable to get an appointment with my GP I went to an open all hours bulk billing GP and the waiting period (2 hours) and subsequent care was sub standard to my normal GP.
Waiting periods for some things are ridiculously long, my daughter is still waiting for a public appointment with an ENT specialist 3 years after having her tonsils out privately - obviously they do not know that she has had them out, I am just interested to see how long they will take to give her an appointment! This is despite her GP writing at the time that she was an urgent case - they told us then that her wait would be at least 2 years (hence paid privately).
However, after a trip to Flinders (public) ED, my husband has found the follow on appointments and care to be excellent.
Around 50% of people in Australia have private health cover, I think the system would be as broke as the NHS if we all stayed in the public sector and with SA health really struggling to make ends meet I can only see the public sector suffering.
I am writing this after a 12 hour night shift (in a private hospital) so it probably makes no sense .... Nite all zzzzz