I can see that the whole island of Ireland belongs to the Irish at its most basic level
Everyone makes this mistake. Tom is confusing "belongs to" with "is run by the government of the nation-state of". The whole of this island does not belong to "the Irish" or to anyone else. For instance, I live in one small bit of it that belongs jointly to me & Vic. That's not a frivolous point.
Framing the dispute in terms of property stolen from "the Irish" by "the British" makes it all look quite cut & dry. It was stolen, so give it back. Except that that's not the type of society any of us live in anymore. When this all started, all the land in Britain was owned by lords. This is a gross oversimplification, sure, but the British Crown stole land from Irish lords and gave it to English and Scots lords, and those lords controlled the lives of their tenants in a very fundamental way that, now, is illegal. Your landlord can no longer tell you where you may work, who you may marry, or force you to fight in his army. If he were to claim your daughter as his rightful property for the purpose of sex, he'd end up in jail.
Nowadays, most of Northern Ireland is indeed owned by Irish people: Northern Irish people. They don't need a change of government in order to be given back their land, as they already own it. Those who don't own land wouldn't own land in a United Irish Republic either. Reunification isn't about giving any land or property back to anyone. What reunification is really about is a change of government, with all that entails: changes in tax law, policing, the school system, the health service, etc. And, even if you think the current Irish Government are a better one than the current British Government (and, right now, I'd probably agree with you), they're subject to the possibility of change at the next election. Nationalists not just of the Irish variety, Scots and Welsh nationalists too firmly believe that Irish people are better served by Irish politicians meeting in a building in Ireland, and that Scottish people are better served by Scottish politicians meeting in a building in Scotland. If this were true, the people of Lambeth, many of whom have a good view of the Houses of Parliament from their bedroom windows, would be rather well off. And the Scottish NHS would have improved since devolution. My aching sides.
Northern Ireland is not owned by "the British"; it is administrated by the British Government. There are, of course, lots of perfectly good arguments for having the province administrated by the Irish Government instead, but none of those arguments have anything to do with giving back stolen property, and such talk of stolen property emotively clouds the issue. Which is why Gerry Adams talks in those terms all the time.
Ownership by people and administration by government are such utterly different things that it's surprising anyone ever confuses them, but people conflate them all the time. And that conflation really screws up one's thinking. In the case of Ireland, let's face it: it's not that big a deal. I'm pro-Union, but, if the island were unified, at least I'd still be in a democracy and I'd be allowed to leave. But people regularly speak about land owned by, for instance, "the Chinese" or "the Syrians" groups of people who don't really get to own much because the tyrannies under which they live don't respect their property rights and barely even respect their right to life. At its most ridiculous, this way of thinking regularly causes people to identify Saddam Hussein with the people of Iraq.
Watch any Robin Hood film. Who represents the people of England? King John?
Tom also makes this other classic mistake:
I mean, just look at the map for heaven's sake.
This popular argument only works on small islands. It justifies not only giving Northern Ireland to the Republic, but also the German annexation of Austria, Belgium, and France; the Chinese invasion of Tibet; Saddam's invasion of Kuwait; and, should they ever feel so inclined, the American take-over of Canada. Funny how the IRA's allies in ETA never invoke it.