I had mine done public, and I got to choose my surgeon (he does public and private and asked me which one I wanted to use, he recommended the public becasue my out of pockets were going to be between $2000 & $10,000 and I have private hospital top cover with no excess, but I am a pensioner and couldn't come up with that out of pocket), as my mastectomy was for cancer I had an up to 30 days wiat (category 1 - must be done within 30 days), my other mastectomy and reconstruction will be done as an elective procedure so I will probabaly have a 1-2 yr wait,but I can do that),
You see the surgeons before you make the appointment for surgery, until you find one you like, who also works in the public system and then go through them.Get a 2nd/ 3rd/ 4tn opinion until you find one you like (ask when you make the appointment, if they do public and private work, so that ou don't waste time), I saw my breast surgeon publically but my port surgeon privately and had both the surgeries done publically (the only difference was that I had to pay to see the port surgeon initially, but the breast surgeon cost me nothing).
That is how you choose your surgeon in the public system. You only don't get a choice if it is emergency surgery. I chose my breast surgeon, port surgeon etc and if I didn't like either I would have chosen another until I found one I liked.
Once you choose a surgeon and agree to surgery, they don't change the surgeon on you.