Its difficult to ascertain as to whether the future will resemble the past few months, especially given that there has been no past flight history on AA in the Australian winter/American summer period. With that said, if you consider the past six months of flight performance, then you most likely will clear on any day of the week even as a non-elite member. There ARE days with heavier loads as well as higher fares and those tend to be Friday-Saturday-Sunday with lower fares on Monday-Tuesday-Wednesday-Thursday. Keep in mind those days are priced competitively with other airlines and may vary and fluctuate dynamically. I am approaching my 12th flight on this route next week and from my experience with booking, upgrades, and total pax, these are the days of the week that have the best chances for upgrade success.

BEST DAYS
SYD-LAX departing Tuesday-Wednesday-Thursday-Friday (arrival same day)
LAX-SYD departing Sunday-Monday-Tuesday-Wednesday (arrival 2 days later)

Considering those that have to work on weekdays/school days
Avoid weekends on SYD-LAX
Avoid Thursday-Friday-Saturday on LAX-SYD

Something else to consider:
It may be easier to follow the school holiday periods for the American summer than working around the Australian winter holiday periods. You can easily search and find information on when school holidays are as they vary by state in Australia and can have a considerable impact on fares and flight loads. The Australian school holidays during summer are the worst as they fall on the Xmas/New Year period and hence why its the most chaotic travel period of the year.

A subscription to ExpertFlyer and use of the ITA Matrix (Google Flights) will be your best tools for researching. You'll be able to view fares in several ways and look at seat maps/fare bucket availability +/- 3 days per search.