I don't work in AWB. I don't work for airservices now. But I did work similar rosters when I did. Everyone liked it because of the time off you got. Finishing early in the morning, having the rest of that day off, then 2 full days off, then not starting til mid afternoon the next is effectively 3.5 days off. We all knew it, as do you. How many guys fit in a round of golf before that 1500 start? That's why no-one wants to finish at 2300 on the last day and start at 0500 on the 1st, as you admitted.
I used to get a couple of hours sleep before the doggo, so I wasn't working "all night". Maybe you should try it.
These days I work a rotating roster still, including nights, so my sleep patterns still aren't the greatest, but they are 12 hour shifts, with an hours traveling either side. My days (and nights) are 14 hours. Also most shift workers do do their nights in cycles, however it is usually a lot more frequently than once a "month or so"
I'm not saying the organization is not understaffed, and that the remaining people are not being overused by management, but having seen things from both sides of the fence, you still don't have it so bad.