So, let's take the Civil Registration Indexes for England & Wales (E&W) and for Ireland as 2 separate examples - both places you would expect to find lots of your Irish kin.
I recommend that you watch Audrey Collin's recent talk to Rootstech 2018 on how the E&W Civil Registration Indexes came to be created and you will appreciate just how easy it was for many errors to have crept into them well before any 20th century digital fingers started flying over keyboards to create the online indexes we all use today. FreeBMD is a great free resource whereby volunteers are transcribing the GRO secondary indexes but did you know that there is also another volunteer project, UKBMD, that is focusing on local registrar indexes from which the GRO indexes were originally compiled (are you keeping up?!) Find My Past (FMP) and Ancestry have then subsequently bought in some of these indexes and / or compiled their own. That's a lot of different indexes starting to proliferate for the same records, so my advice is, if you can't find what you're hoping to find in one index, try out another one.
The free Irish government website, Irish Genealogy, has their own indexes to the Irish Civil Registration records, whilst I believe both FMP and Ancestry both bought in the index compiled by FamilySearch for the same records. But it would appear from a recent story on Claire Santry's Irish Genealogy News blog that FMP have now created their own indexes to Irish civil births and marriages but there does seem to be some quality control issues.
Incidentally, I gather that perhaps somewhat unusually, FMP and Ancestry actually collaborated on the creation of new indexes to the Irish Roman Catholic records released by the National Library of Ireland - presumably to keep costs down rather than try to race each other to be first to publish? So, if you are searching these records in either of these sites, I presume the results should be the same? I haven't tested this one out for myself yet.
So, the moral of the tale is don't rely on just one index that happens to be accessible via your favourite free or subscription site. If you want to be thorough and "reasonably exhaustive" in your research, then use as many different indexes as possible in order to compile your candidates.