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it actually sounds like we disagree on a scenario here.

Normally, an Irish citizen with US PR can fly into Canada on just the Irish passport + US green card, no ETA required. The airline wouldn't know said person is also Canadian, and so would let this person onto the airplane.
I just realized that I made a mistake here, which may have also lead to some confusion regarding the disagreement. I don't recall now why I thought that green card holders didn't require ETAs, as per https://tiac-aitc.ca/Electronic_Travel_Authorization.html and https://rjimmigrationlaw.com/resour...sident-travel-to-canada-your-essential-guide/ they actually do require ETAs.

So no chance if you have a passport that is already visa exempt as you need an ETA to fly in with or without the green card. What's different is if you have a visa-requiring passport and a green card, then you can use the green card to fly in instead of getting a visa, but you then need an ETA too.

Thus if an ETA is required and you somehow have both the US green card and Canadian PR, then presumably you'd show up as a Canadian PR in the system and it would still reject you. So actually a green card doesn't change anything and isn't of any help in flying into Canada.