My guess is that the paragraph UAFlyer cites is a fossil that was incompletely edited some time ago. All lounges in ATH are post-BC check and pre-security and they're post-passport for extra-Schengen flights.

As for the A3 check-in counter, the airline has abolished check-in counters in most airports: self CI (home or machine) are the rule, even outside of Greece, perhaps with the exception of business class and A*G (counters are still there in ATH). This only partly answers the question of UAFlyer...

As for the choice between the A3 and the OA lounges, this is purely a matter of choice. I personally prefer the former, but there are many who disagree with me. Since you (RudderDucky) have three hours time, check for yourself. But don't forget to ask for a priority sticker at either the A3 lounge or the check-in counter (if you use it), although at that late time of departure, security check should be rather fast.