There are alternatives offered in LAX too, namely the *A J and F lounges in TBIT that anyone could walk to if they wanted, and which are run on behalf of the entire alliance.

If you want to argue that *A policies say *G get into all J lounges operated by a *A airline, go for it. You're on decent ground, and there would be a lot of people that agree with you. But when you accept that UA can exclude *G from the Polaris lounge, which is a J lounge, then it's tough to swallow you also arguing that other *A carriers shouldn't be allowed to customize their access policies too.