I don't know what to tell you. The flights were on one ticket and checked through, something EI explicitly doesn't do for point to point bookings. Not sure why you'd tell me I don't know what I'm talking about when I was tramping about the missed bags room trying to find my suitcase which also didn't connect after being tagged through to Donegal. Do you need to make a dummy booking on an OTA to convince yourself that it can be done?
FWIW the tickets are more expensive when booked as above rather than point to point, probably because it includes insurance for EI against missed connections. But when you want the convenience of through checkin as I did I was happy to pay a bit extra.
Don't want to have a big debate but there are plenty of sites where you could for example book CDG-CFN and have it issued as a single ticket but EI could just as easily refuse to accept it as a single one. Any airport agent will check bags through to x destination if you show the onward tickets if seperate and the same airline in most cases.