You mention finding the name Patrick Moss in Griffiths Valuation. Griffiths, for Co Tyrone, was compiled in 1859. Since your family was in the US by 1851, none of the entries there can be your immediate family, though they might be relations.
I searched Ancestry’s Roman Catholic baptism and marriage records for Patrick & Ellen’s marriage and for baptisms of their 3 or 4 children born in Ireland. I did not find them. That may because the events took place in a parish that does not have records for the 1840s (Many don’t).
Returning to Griffiths Valuation, though your immediate family can’t be in it, it nevertheless gives an indication of where in Tyrone the surname was to be found. I noticed there was a Patrick Moss in Garvagh townland, in Termonamongan parish and another Freughmore, in Drumragh. The one in Garvagh is in 4 times because he appears to have more than 1 piece of land in Garvagh. Irish families often use the same names, over and over again. John & Ellen are fairly common in Ireland but Alice slightly less so, especially in Catholic families. So I was interested to note that an Alice Moss from Garvagh got married in 1879 (father was Edward). See below.
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1879/11056/8044617.pdf
This might be a total coincidence and completely unconnected with your family but the existence of a Patrick & Alice Moss in Garvagh makes me wonder if there might be a connection. No McGinty households in Griffiths in Termonamongan at all. However there was a McGlinchy family in Garvagh. Could that be Ellen's maiden name, which got garbled by whoever was the informant for the death certificate? There were several Moss households in Garvagh in Griffiths, probably all related.
Alice Moss from Garvagh married Francis Curran from Carndreen. Here they are in the 1901 census:
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Maghernageeragh/Carndreen/1723745/
Garvagh is a rural area a few miles north of Killeter. Today it’s on the modern Carn Rd, half a mile or so before the border with Co. Donegal. Population in 1901 was 170 people (nearly all engaged in agriculture in one form or another). There were 25 homes including 6 Moss households. One of which included another Alice. (There were only 2 people named Alice Moss in the county in that census so it's not a common name).
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Maghernageeragh/Garvagh/
Termonamongan’s RC parish records don’t start till 1863 so if your family came from that parish, it would explain why they can’t be found in any on-line records.
Several trees on Ancestry show your family as coming from Garvagh, and so others seem to have come to the same conclusion.
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).
