You're missing the fact that AA can only be a viable competitor in this market with the ability to tap into QF's network.
If AA and QF are competing in the market instead, then QF has no reason to feed AA flights. Therefore the AA flights become unviable. Therefore AA exits the market. Therefore QF restarts a 4th frequency SYD-LAX and removes SFO service. Which reduces competition in the market.

If BA & AA didn't have a trans-Atlantic JV, do you think AA would have anywhere near as much emphasis on LHR as it does today?