Why am I unable to find any record of a death certificate, even when entering the dates that you gave me.
Where are you searching? Irish Genealogy only covers N.I. counties up to 1921. For later dates you need to search GRONI's database. Certificates not online but you can view them for a few credits after you open an account (free).
https://geni.nidirect.gov.uk/D/1953/148/1018/57/225 Arthur Nelson 7th Sept.1953 84 Lisburn(pre-1973 Q4)
D/1954/148/1018/57/427 Maria Nelson 4th June 1954 82 Lisburn(pre-1973 Q4)
Also, you have named sites I have never remotely heard of, but which are superb sources of information. Is there any 'list' available that can give us less experts a wee help?
Lots posted in Ireland Resources, Antrim Resources, pinned to top of some boards, etc. (some links outdated but I am not able to make changes and update them anymore)