As the OP's flight was on one ticket, then nope, he shouldn't have had to pay the ex-UK departure tax. Avoiding that tax is exactly why some FTers that have the time to do so buy an el cheapo, separate ticket across the channel & then on one ticket go from there, thru LHR, to the US or wherever....as sometimes, that el cheapo flight is a LOT less than what the APD would be on an upgraded ex-LHR flight.
If the GA is saying he/she has to collect it, then that's an error on the GA's part. Whether or not the GA SHOULD have the hardware/software to collect it is another story. It can be done via ticket counter & lounge personnel, and I'm guessing that's likely the response that a GA who would know that would say. That's possibly what AA will say as well.
Some of the really good AA GA's that post on this board would likely say they've got enough to handle as it is getting flights out, and in fairness to them, I'd say that---having flown ex-LHR many times this year---the boarding area is usually quite a zoo (not the GA's fault, just the plethora of demanding pax crowding around).
Yep, a quick check back w/ the Flagship Lounge MIGHT have helped, and/or so could a call to the EXP desk (and those agents in Dublin are always quite helpful), but hindsight's always 20/20.
It will be interesting to hear what AA's resopnse is. There are threads here about the inconsistency of ground staff at Heathrow, and perhaps this is yet another example of that, needing AA's attention. Or, perhaps it's just an honest mistake by the GA in the middle of a boarding process.