Now to the substantive question:
As I have noted in the biography of your Patrick Brien, in Killurney in Griffith's Valuation (25 Oct 1850), Patrick Bryan, the only Bryan or variant in the townland, occupied (no. 3) a house with rateable annual valuation of 10 shillings on 4a 3r 1p.
A few months earlier, there are listings for Patrick Brien in two townlands in Youghalarra parish in Griffith's Valuation (14 May 1850), Ballymoylan/Ballymoylin and Barbaha:
https://www.failteromhat.com/griffiths/tipperary/youghalarra.htm
These comprised Patrick Brien (Tone) and three plain Patrick Briens occupying land and buildings in Ballymoylan and a landless Patrick Brien in Barbaha:
https://www.askaboutireland.ie/griffith-valuation/index.xml?action=doNameSearch&Submit.x=47&Submit.y=14&familyname=brien&firstname=patrick&countyname=TIPPERARY%2C+NORTH+RIDING&parishname=YOUGHALARRA
There were still four (O')Brien households in Ballymoylin in 1901:
http://www.census.nationalarchives.ie/pages/1901/Tipperary/Youghalarra/Ballymoylin/
but none left in Barbaha:
http://www.census.nationalarchives.ie/pages/1901/Tipperary/Youghalarra/Barbaha/
Ballymoylan/Ballymoylin is 90km from Killurney:
https://goo.gl/maps/8zjetYukjTst4kRZ9
Mrs. Brien of Youghalarra was Anne or Nancy in the baptism records of all of her children.
Mrs. Brien of Killurney was Anastatia in the baptism records of all of her children.
These are almost certainly just parallel families, and almost certainly NOT a single family hopping back and forth between the two ends of the county. Even the Templederry and Youghalarra families are likely just parallel families.
I found another child in the Killurney family (a second John/Johannes baptised on 14 Feb 1847), who was picked up by ancestry.com's indexers but may have been misread by rootsireland.ie's indexers.
The failure to find marriage records is most likely down to the brides being from nearby parishes for which the marriage registers have not survived, or else down to transcription or indexing or translation errors. It is important to bear in mind that survival dates for parish registers vary dramatically from one parish to the next.
The Catholic parish currently described as Kilsheelan & Kilcash and described by the NLI as Gambonsfield and Kilcash is probably coterminous with the two civil parishes of Kilsheelan and Temple-etney, but you would need a local expert from that end of the county to confirm my conjecture. By coincidence, my own second cousin Fr Paul Waldron is actually the current parish priest of Carrick-on-Suir, two parishes to the east!