Wow. I didn’t know that things are this way. I am a recent immigrant living close to the city and always seem to be able to see a GP on the same day. Is that because I have private insurance and I pay out of pocket anyways? If I were a citizen, I wouldn’t be able to go to those places (at least for “free healthcare”), and will have the same wait times? Or is it geography dependent with rural Victoria having issues and Melbourne city being well covered?

It depends on the location and it depends on what services you're after. If you have private healthcare insurance, you get to skip the wait times at hospitals and get a choice of public or private hospitals. For clinics, it's a different story and can get quite detailed. The private insurance can help you with the payment there but not wait times.

For some regional and rural locations, the wait times can be better or can be worse than metro depending on the service.

By the way, I also pay out of pocket on top of the medicare rebate so my experience is not with bulk billing clinics. When you get access to medicare, you'd probably still need to pay out of pocket on top of the rebate as bulk billing clinics have all but disappeared. Recent government incentives aim to bring them back but with cost of living increases I doubt that'll work.

With seeing a doctor we have two main systems that you can use and each will have a different waiting time. Bulk-billing and the fully public option has longer waiting times because there aren’t enough clinics/specialists or doctors, The reasons for this are complex but they stem from an unwillingness from prior governments to raise the amount the government pays for each service to adequately to support this system meaning less doctors and practices being willing to support it.

You’ve then got practices/specialists etc… that charge copays and they tend to have less waiting times because less people are willing to pay copays. A lot of these practices will also do outright private billing which is what you’re experiencing.