Slainte!

Mick Dunne
Philadelphia Shamrocks

But many continue to have the village or the neighborhood or parish as the defining reality, and the GAA plays a huge role in this, thank God.

The GAA was a big part of the Irish response to the famine, along with the Land League and the revival of Gaelic and the formation of organized efforts to build a separate cultural and psychological identity separate from that of the UK and the inferior position of the Irish in the UK. This cultural revival occurred alongside the formation of organizations that would be ready for complete Irish independence from Britain, an independence to be attained through either UK constitutional means (as with Canada) or through armed insurgency (IRB/IRA/Sinn Fein).

During the brief Irish Civil War that followed Irish independence, the GAA staged games that drew players drawn from both sides, who were pledged to keep politics out of the GAA (except for Irish independence).

If you want to go into ancient roots of the game, look to head hunting. The ancient Irish used the shrunken heads of enemies as the thing to wack around. In fact the oldest church in use in Ireland in Clare has a portal decorated with heads, evocative of the old sacred Celtic status of severed heads. This head hunting phenomenon can also be seen in France in the legend of St Denis.

Congratulations on learning to move beyond Irish-Americanism to discover Ireland. I too am an American, with parents born in Ireland but I avoid the entire Irish-American thing. Not my culture.
The GAA