I just arrived in Athens this morning on AA, starting in Raleigh and connecting in Chicago and I want to write this while it is still top of mind (though admittedly jet lagged).
Raleigh - only asked by gate agent to scan my passport - no other documents asked for.
Chicago - everyone had to stand in line at the gate to have the following docs verified: their passport, vaccination card/negative covid test, and a print out of the information inputted online showing that you'd filled out the request for the passenger locator form (PLF). Since our flight left at 1:30 Friday afternoon and arrived in Athens 7:30 Saturday morning, we would not have the actual PLF until we landed. I actually don't know what all counted as proof that you'd filled out the form, but I did witness several folks get told they could not get in the second line to board the plane bc they could not show proof that they'd filled out the form. This first line was very slow, with only 2 gate agents frantically trying to check everyone's documents. Once the documents were checked, the agent stamped my boarding pass with "ORD DOCS OK" in red ink. This allowed me to get in the second line. In the second line they were taking a photo of each person before letting them on the gangway, but about 2/3 of the way through boarding, they abandoned this practice, I'm guessing bc it was taking too long? I ended up in the line that didn't take my photo. Only my boarding pass was scanned. No other documents were re-verified.
Athens - first waited in a line to meet with two plain clothes men (no uniforms, no name tags) checking for passport, cdc card, and PLF QR code on my phone. It was handy to have internet access while I waited in line! I'm not sure what someone would do if they couldn't connect to the internet to show the QR code (which had come through while we were in flight).
Second, waited in a line to get my passport stamped. To the far left was a line for - I think - people that were pulled for random covid testing. Folks were accidentally getting in that line bc they were reading the big official signs that said "non-EU passport". That is misleading!
For both of these lines, our flight was the only one being processed.
There were 5 agents stamping passports. The agent did not ask to see anything except my passport, and did not ask any questions. That line moved quickly.
From touchdown in Athens to clearing boarder control was about 45 min easy peasy. I was on the back third of the plane, which probably slowed my thru-time.
I hope this helps!