So you are going to fly in a tube for 14 hours or so, from one continent to another, on a flight that goes once a day.
AND you are trying to cut it fine for check in (or that might just be my perception)
I agree that the website should publish a cut off time,
However you should allow time for delays in traffic, delays in finding the VA desks in a huge terminal, delays once you get checked in, with security and maybe a long walk to the gate, and you will want a moment to go to the bathroom, get a coffee or a snack, make a few last minute phone calls or send some emails, or just take a selfie for FB
Now if you check on just on the cut off, you may make the flight by running, you may be the last to run onto the aircraft BUT will your bags get the same treatement.
Get to SYD and stand and wait at the luggage belt. Wait some more, watch everyone else get their bags - the belt stops you bags still not there - guess what they are still at LAX - go fill out a luggage report etc.
So much easier to get to the airport at T120 and chill