I think I have a full conclusion to this thread.
I'll write my methods so that others can either double-check, and/or learn from my difficulties.
My objective was to find the names, dates of birth, and dates of death of my maternal grandfather's grandparents (my 2nd great-grandparents).
I started with my maternal grandfather's parents. Details was sparse, due to estrangement of the grandfather and his parents - but I had their names, dates of death, and location of birth for the great-grandfather.
Unfortunately, I could not find the marriage records of my great-parents - which I now know to be 1910 - so there was always going to be some guesswork involved.
I got the date of birth for the great-grandmother - based on her death record - however, I identified the wrong individual (see my previous post). I eventually found the correct person and got the names of the maternal set of 2nd great-grandparents (I haven't got their dates yet, but that's another story).
I traced the paternal set of 2nd great-grandparents to Clonakilty, based on references from distant cousins; I got their names from a well-informed tree on ancestry. The great-grandfather's birth record connects them, at this location, and I found a number of other children born to this couple - then the trail went cold - as outlined in my original post.
I returned to civilrecords.irishgenealogy.ie in the hope that I could find something I had missed - and I found it: four additional children. They were not transcribed by parent name, and I found them by opening each birth record for Murphy in Clonakilty between 1880 and 1900.
Now I had a list of children - in date order - with the youngest having a relatively uncommon name: Julia. I entered these names into the 1901 census and found a household in Kilmaloda, Clonakilty - less than 10km from their last known location (Ballinrougher):
https://www.census.nationalarchives.ie/pages/1901/Cork/Kilmaloda/Ahaliskey/1097602/
The 1911 census followed:
https://www.census.nationalarchives.ie/pages/1911/Cork/Kilmaloda_West/Ahalisky/381879/
I'm not sure why John is listed, given that I know he was in Belmullet in 1911

Finally, I got their death records, matching the Ahalisky location, and the informant names as their children:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1908/05475/4532823.pdf I previously thought Ellen had died early - but that was another Ellen Murphy
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1926/04992/4361740.pdfI think the moral of the story is: don't rely on digital transcripts as 100% accurate, and be prepared to sift through if the records are available as scans.