would the Irish model make for a useful case study when considering
the various advantages and challenges of such an arrangement?
No.
Ireland and Northern Ireland are very similar
One of the biggest problem with using the border between the Republic of Ireland and Northern Ireland as a model of an "open borders policy" is that, despite the tyranny of small differences (mostly Catholic v. Protestant distinctions), the two populations have so much in common.
They speak the same language. They drink the same kinds of whisky and beer (although they prefer different brands). They eat the same kinds of foods. They are in the same climate zone. They are both predominantly Western Christian. They share most of the same history. They are both quite similar looking white people. There aren't huge differences between them in affluence or legal systems. They drive on the same side of the road. (The Republic of Ireland has a much higher per capita GDP than Northern Ireland, but this discrepancy is mostly a quirk of the way GDP accounting works in the case of a county that is, like the Republic of Ireland, basically a big tax shelter for foreign multi-national corporations and rich people who don't really live there or spend much money there.)
One might compare the degree of ethnic difference between Northern Ireland and the Republic of Ireland to the degree of ethnic difference between heavily Catholic Rhode Island, and adjacent historically Protestant Connecticut, in the United States, which have also had open borders between them for a very long time.
You can focus on the small differences, but political concern about "open borders" paradigmatically, focuses on different situations. Situations where there are huge differences in affluence and education, where there are big cultural, ethnic, and racial differences, where migrants flow overwhelmingly in one direction, where the migrants aren't familiar with the legal system of the country to which they are migrating, where there are different norms about acceptable and unacceptable behavior, and where the religious differences are sometimes much, much greater.
There is little migration across this open border
Ireland and Northern Ireland, after a long period of free immigration (including all of the U.K.'s E.U. membership pre-Brexit and under treaties before then, and the long period when they were part of the same country prior to 1922), and no real language barriers, is very well sorted. People who want to be in the Republican of Ireland live there. People who want to live in Northern Ireland live there.
There is little pent up demand to relocate from one to the other. In contrast, the early phase of an open borders regime contemplates a deluge of pent up migration from the closed borders era.
Ireland and Northern Ireland are both mostly sources of migration to other countries, and destinations from other countries, neither of which mostly involve open borders.
Ireland and Northern Ireland also have historically been net sources of immigration to other places (both before Irish Independence and afterwards, on both sides of that border), rather than destinations, which is not the paradigmatic open borders scenario. The total population on the island of Ireland in 1841 was 8.2 million (according to census figures which may have undercounted its population), and may have reached a peak population of 8.5 million by 1844. Including today’s Northern Ireland population, the island now has 6.9 million people (as of 2022 when I last traveled there). Basically, even now, about 180 years later, Ireland's population has still not recovered from the population that it lost in the Great Potato Famine of 1845-1849 and its aftermath.
The Republic of Ireland has actually recently welcomes a lot of immigrants from many places around the world and has a similar percentage of foreign born people to the United States. But, this is very recent. Again, as of 2022, more than 17% of the population of the Republic of Ireland is foreign born (higher in Dublin, lower elsewhere) compared to 15% in the U.S. But, 46% of the foreign born residents have lived in Ireland less than five years, so immigration into the Republic of Ireland is mostly very recent indeed. By comparison, the U.S. had a foreign born population of about 5% in 1970 at its low point, which has grown relatively gradually over 55 years to its current 15%.
However, while Ireland has welcomed people from other countries (including many Ukrainian refugees), this hasn't been the result of a genuine open borders policy, unlike the migration back and forth across the border with Northern Ireland.
Ireland's history of foreign land ownership is confounding
One other quirk about Ireland that makes it an atypical example is that as of 1922, a century ago, when the Republic of Ireland formally gained independence, only 5% of the land in Ireland was owned by Irish people. This had profound economic, legal, and political implications that could cofound any effort to determine how much this rather than an open borders policy influences anything that is observed that might be attributable to the open borders policy.
Puerto Rico is a better example from which to generalize
A better paradigm for the kind of "open borders" that people who are concerned about them are interested in, might be the sustained open borders between the Commonwealth of Puerto Rico and the continental United States, since Puerto Ricans gained U.S. citizenship status in 1917 (and in most cases, earlier). Puerto Rico has a great deal in common with other Spanish speaking territories and countries in the Caribbean and along the Gulf of Mexico's South American and Central American coast.
But, unlike all of those other countries, it has had more than a century of open borders with the rest of the U.S. Roughly half of people born in Puerto Rico who are alive today have migrated to the mainland (especially people of peak working age), and this continuous migration away from Puerto Rico to the mainland United States is ongoing, despite having more than a century to stabilize.
The E.U. is also a better example.
The E.U. is a case of a region of very different countries having open borders even more recently, and provides the added benefit of allowing an observer to look at the impact of the expansion of the E.U.'s border free region over time that can test if hypotheses about the cause of certain effects are really due to open borders or have some other cause that just happens to coincide in time.