The rules are definitely unclear, but the way I read it the lack of clarity actually leans toward access being granted, not against. The specific and entire paragraph is (emphasis in original):
So what it's saying is that first and business class passengers on domestic flights don't get access to AA operated lounges, but no mention is made to non-AA lounges at all. Read strictly, this rule only says that such passengers don't get access to the Admirals Club or Flagship Lounge. Since the actual rule is that "customers flying First or Business Class have access to the equivalent class of lounge regardless of their frequent flyer status", I read this specific restriction to that very broad access entitlement to apply only to Admirals Clubs and Flagship Lounges, leaving the door wide open to access to non-AA-operated oneworld lounges such as the ones at TBIT. The paragraph then goes on to say that first and business passengers on JFK-LAX, JFK-SFO and MIA-LAX
do get lounge access, which seems to modify the above restriction (thus removing the limit from AA lounge access for them), and doesn't make any further restriction on what kind of lounge they can access, which by my reading leaves them open to all oneworld lounges, both AA and non-AA.
The way it's worded (and the way I'm reading it), it's a bit like saying: "Car drivers cannot use BP stations on highways in Anytown; If you drive your car on Anytown Highway 100, you are eligible to use petrol stations on this road." The first clause seems to limit car drivers from accessing BP stations without mentioning other petrol station brands at all, while the second clause indicates that petrol stations can be used on Highway 100 without any restriction on station operator at all.
At the end of the day, this rule is so poorly written that I can't find any reading of it that accurately describes 1) what's actually happening on the ground, or 2) what the rules are meant to say. So maybe this entire discussion is academic.
