Since the ETA is tied to your passport, the only real way to check if you have one is to show your passport to the airline.
But as you said, this is not immigration. It is the airline checking you have the right documents.
What happens is that when you check in, your passport information is sent to the UK government to check you have your ETA. (Of course, British and Irish subjects are exempt but the only way the airline would know this is to look at passports.)
I haven't traveled between the UK and Ireland since ETA has taken affect so I understand this change.
I had an issue a few months ago in Dusseldorf. When I went to drop luggage, it wouldn't let me and said I had to see an agent. He checked and said I didn't have an ETA. I said I did. He fiddled around with him computer and then said, "There it is." The computer, he said, was causing problems that day.
On a BA flight a few weeks ago, I had to show my email confirmation because the computer didn't have my ETA information. It's possible because my original AA flight was canceled and I was put on a BA flight.