You are SO wrong about that. This was one of the key elements of the GFA: that people born in NI could choose whether to BE (not just identify as) British or Irish. And that isn't just reflected in the fact that half of the population has chosen never to hold a British passport. Until recently of course the other half ONLY held British passports, but that all changed with Brexit - now they're all merrily applying for Irish passports, including Ian Paisley Jr.
There was a landmark court case while the UK was still in the EU, of a woman from NI who demanded to be considered as Irish, NOT British. She had been told by the Home Office that her American husband could not live in NI with her, because they failed the stringent rules imposed by the UK on their own citizens for bringing in non-EU spouses. She was invoking the unconditional right of EU citizens to live with their spouse in an EU country OTHER THAN THEIR OWN ( because in your own country, that country's laws supersede those of the EU).
She argued in court, invoking the GFA, that although she was born in NI, as were here parents etc, they were IRISH, not British, so effectively living in another EU country to their home country. So the same rules should apply to her as if she'd been born in Ireland and later moved to NI. And she won.
From then on, the UK recognises that anyone in NI who sees themselves as purely Irish is so. This means that these people are also protected under the Withdrawal Agreement, just like any other EU citizens living in the UK before Brexit - which doesn't apply to Brits. There's a list of advantages, such as continued entitlement to EHIC (which covers more countries than the new British GHIC, which is all British citizens are entitled to).
But the main point is the principle. The very many people in NI who don't see themselves as British, only Irish, are Irish. Period.
Going back to when this was first established, in the 1920s (even though the UK didn't respect it in many ways until the GFA), this is of course the reason for the Common Travel Area between the UK and Ireland. Irish people in the UK have exactly the same right as Brits, including to vote. And vice versa. It has to be that way, because a very large proportion of the population of NI is Irish, not British.
End of History lesson :)