These days many descendants of such Irish prefer to be known as Scots-Irish ~ but they're not!
There are a number of reasons why the Catholic church wasn't very well organized in the United States in that period (1800 to 1854), but most of it gets down to the fact that there simply weren't very many Catholics to "organize"! Newcomers were on their own. The Baptists were in strength and welcomed then with open arms (as did 9 synods of Lutherans, etc).
In addition to the Scots-Irish, there were Catholics whose property had been confiscated and who lived by renting and farming small pieces of land. The Irish immigrants were poor, so many of them could not afford to pay for their passage to the United States, so they came to the US as indentured servants. This meant that, in return for their passage, they would agree to work for someone for a certain number of years, usually seven year. Many former indentured servants settled in the Appalachian Mountains, where they farmed small pieces land with a few animals, growing perhaps a little tobacco and cotton, some of which they sold for cash. While they were not wealthy, it was a very good life for them, compared to what they would have had in Ireland.
By 1776, a quarter of a million Irish people, mainly Protestants, had immigrated to America. In general, they were happy in America, because they could practice their own religion, farm their own land, express their opinions freely, and participant in politics.
When the American Revolution came, the Irish were natural revolutionaries, because they already had a great hatred of the British. They played an important role in the revolution, disproportionate to their numbers in the population as a whole, as soldiers, nurses, and officers. It has been estimated that the US military during the Revolutionary War was up to one-third Irish and Irish American. The "Famine Irish" Up to the 1820s, the typical Irish immigrant was an Ulster protestant, though a few Catholics had been immigrating from the beginning. However, beginning in the late 1820s and early 1830s, more and more Catholics from Ireland immigrated. By the 1830s, the typical Irish immigrant was a Catholic, and was usually very poor, even in comparison to earlier Irish immigrants.
My parental gene pool as a 7th generation Mississippian has seen no new immigrants since the 1750s from Ulster.