I'm afraid it's always the responsibility of the passenger to have the correct documentation to fly - it doesn't matter if it's a visa, US ESTA, Covid declaration, ticketed reservation, passport or ID card.
The airlines will check. Timatic, which is what they rely on, has this for entry to Greece from France (assuming you have a French/EU passport):
Originally Posted by Timatic
Warning:
Passengers are subject to medical screening and quarantine for 1 or 14 days.
Passengers must complete a "Passenger Locator Form(PLF)" 24 hours before arrival at
https://travel.gov.gr/ . A QR code generated from the completed form must be presented upon arrival.
By your own admission, you did not have a correctly completed form. Had you just shown the agent the QR code the likelihood is that they wouldn't even have noticed, and I equally suspect the Greek authorities would just have processed you if you'd maintained the same silence the other end - but as soon as you mentioned the problem then they were always going to take the most cautious line, which was to deny you boarding. I imagine the possibility of you raising alarm bells in Geece by saying "This isn't my name" to the authorities there wouldn't have helped.
Contrary to your belief, and that of the other passengers, airlines are fined large sums of money if they allow a passenger to travel who is subsequently refused for not having the correct documentation or permission to enter the country. In the circumstances, the airline was perfectly within its rights to refuse to carry you.
I will say A3 is not the only airline denying boarding on these grounds - there are accounts of BA doing the same at Heathrow, albeit they seem to be because the passenger has made no attempt to complete the form.
You need to chalk this up to experience and know that in future it's better not to voluntarily raise anything that you believe won't be an issue in reality.