I really think that the Irish origin is only part of the history of this name. I think it is just as much a transfer to given name status of the English surname, which has multiple origins, including from an Old French word "barri" meaning "rampart", a Scottish place name, and a form of Welsh "ap Harry", "son of Harry". There are 74 men and boys with Barry as a given name in the 1841 census of England. 20 of them were born in Ireland, but most of the rest have no obvious connections with Ireland. I do think from looking at U.S. Census records that the Irish origin was the most important. I just don't think it is the only origin of Barry as a given name.
Possibly derived from either of two Irish words, one meaning "spear" and the other meaning "fair-haired" (ie. blond).Rarely used as a diminutive of the name Bartholomew; and I suspect that would be because of Barry Allen, the most well-known of the DC Comics heroes to take the codename of the Flash (who does indeed have the former as a proper first name, though it's almost never noted).
I do think from looking at U.S. Census records that the Irish origin was the most important. I just don't think it is the only origin of Barry as a given name.