I would imagine there's relatively few people on an AA flight that present themselves at the BA lounge.

AA now operates out of the Harvey Milk Terminal, T1. It's pretty much brand new, so everything works - security is quick (at least for PreCheck, ime), the terminal is bright and airy, and the AA lounge well situated for the gates - and for a normal club as opposed to a Flagship it's quite decent with a reasonable range of food and very helpful agents, coupled with good views over the tarmac.

In the normal course of events, most passengers wouldn't turn up particularly early for a domestic flight. If I'm ever there two hours before I'm generally cursing myself for allowing too much padding time for the BART from downtown. So the ideal for me, and most others, is to cut it to about 1h30 at the outside.

It's a reasonable schlep from the side of the International Terminal BA uses over to T1, and you need to clear security again. If you arrive at the airport less than 2 hours before, it's neither worthwhile nor viable. So I can see there are occasions, such as a long transit, where it might be possible to visit the BA lounge if it's open (and not operating as PP), but it's not going to be particularly common.

Should every agent get it right? Yes, probably, in a BA lounge with a BA Gold. But the reality is that you're talking about being in a very small subset of passengers and I'm not sure an agent would see this scenario even once a day when they're processing hundreds of people.