Have you found any other Michael Bradys with a father Philip from Terenure?
In the vicinity (Terenure, Rathfarnham, etc) yes. There are two with parents Philip Brady & Marianne Lynam:
15 Jan 1870 (least likely)
27 Dec 1871 (most likely)
These two also have a Michael baptised 31 Aug 1862.
and by Philip Brady and Catherine Reilly:
27 Sep 1870
(I mention the latter because I think he'll be important in finding out details of Michael's father - I have a strong hunch they're cousins, as are the two Philips).
Though another angle was mentioned in an earlier post by shanreagh - if he wasn't even born in Dublin, then I'd be forcing "loosely fitting" pieces together when the real person could be elsewhere.
I think my reluctance to pull the trigger is just based on all of the other evidence being VERY strongly tied, whereas the link I have between Michael and parents Philip/Mary is very loose. I guess I'd rather spend time trying to get all the pieces, as there is SOOOOO much misinformation from other trees on Ancestry. And also the fact that 'Mary' in other parts of my tree has been VERY interchangeable, sometimes even a complete swap of first and middle names!
Philip, in July 1903, was living at St Patricks House, and elderly home:
http://www.rootschat.com/links/01sca/The census from two years prior:
http://www.census.nationalarchives.ie/pages/1901/Dublin/Ushers_Quay/From_O_Connell_Road_to_Valencia_Circular_Road_South_Side_/1303121/Which is another angle i'm coming at, to see if (rather than getting Michael's details directly) I can get Philip and Mary and perhaps details of them and work down.
Though even then, a death record from 1903 that matches St Patricks house (and based on other records, is AROUND the same likely time of death) also puts him at 80 years old, two years after the census that had him at 80:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1903/05647/4589479.pdf