How come these agents couldn't something this simple right from day 1?

It's not that hard at all!

Non-AA OWE flying on any OW flight: check

AA EXP flying intl outside of NAM except for MEX: check

ANY OW 3 class F ticket (includes JFK/MIA 3 class F on AA): check

Arriving from intl 3 class F with any onward OW flight: check

Any intl arriving OWE with any onward OW flight: check

I think that's about it, just those five rules. Of course, in my experience, most lounge agents are too lazy to read the complete rules and they just stop reading after they see "AA EXP Domestic no access" and they just refuse to read on for the exceptions. Then again, anyone who is more conscientious is probably not working that job...