Note this thread is about the vaccine roll-out in Australia, not just in NSW or in Sydney. While your statement maybe true for NSW or Sydney, it most certainly is not correct and extremely misleading for people who reside in other states/cities. Please don't lump all Australians under the process that operates in NSW.
My own circumstances, living in South-East Queensland, do not match your stated "facts". I am in the 50-59 category of group 2. I became eligible for vaccination in May as you stated. However, being eligible and being available are two completely different things. I had a GP booking to receive first dose of AZ in June. 2 days before that appointment the recommendation for under 60s to receive Pfizer was announced and my GP booking was changed from a AZ vaccination to a general check-up with my GP. My GP continues to only offer AZ generally to over 60s, and only to under 60s if there is a medical reason not to wait for Pfizer - I had that discussion.
So I immediately registered with Qld Health (Register your interest to receive a vaccine). According to the Qld Health on-line registration system I am eligible for Pfizer, but they will not have any availability until October, at which time they will contact me with a URL link to make a booking. I expect to be "double-dipped" with Pfizer some time in November.
So please stop the false rhetoric that 50-59 year olds have had 3 full months to book and get a jab. I am not even able to make a booking yet, let alone receive a vaccination. Some have had that opportunity, but certainly not all as you have incorrectly stated so many times in this thread.