Sean,
You ask whether it’s common not to find any records. The answer is yes. Problem you face is that many RC parishes didn’t have records for the years you are interested in (1849 and earlier). You can use this site to check, parish by parish, when the records start for each.
https://www.johngrenham.com/browse/counties/rcmaps/tyronerc.php#maps/
Taking as an example, Termonamongan, you’ll see their records only start in 1863. Badoney Upper & Lower both start in 1866. So if any of those is where the family lived then there are no records to search. No easy way round that.
Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).