The Irish Tricolour flag represents the country of the Republic of Ireland, 26 counties of the island, which in fact doesn’t include the 6 counties that St Patrick mainly operated in. In fact, it can be quite offensive to many people living in the place St Patrick mainly operated in, due to a civil conflict.
St Patrick also has his own flag - a red saltire on a white background - which deliberately represents him.
Making St Patrick’s day a festival of Irish Nationalism specific to the Republic of Ireland makes zero sense, and using a flag which only represents part of the island, especially since many Americans are also descended from Ulster Scots in Northern Ireland.
St Patrick does have connection to Ireland, due to his mission affecting the whole island, and he was kind of patron saint of the whole place before the split anyway.