Edit: Funny I replied the answer to "Obviously none of them spoke the language, and he'd ask why not? Great question."
Why did you just move on? You should be happy to have your 'great question' belittling my family answered. It's because of death, and survival, and scraping by to survive, lots of pieces got lost. That was your ownage. That our families were broken people just surviving and sometimes language was one the the pieces we lost. Pieces we are excited to maybe explore when we visit europe, (until we run into people like you). I have an old family bible with Gaelic that my family wrote in it. But that isn't a connection, right?
The problem we all see is that you're saying that you are Irish. If you weren't born in Ireland, your parents weren't born in Ireland, you don't speak Irish, you don't pay taxes in Ireland, you can't vote in Irish elections, you wouldn't join the Irish military, you don't understand Irish culture, or know anything about Irish history, then in what way are you Irish?
You're not. But you have redefined "Irish" to mean something else. And that's what pisses people off. There are actual Irish people out there. Invent your own identity.
And Americans are claiming ethnic ancestry - not national ancestry. And many Irish migrated out of Ireland, when it was not a country - and paying taxes is irrelevant in this gatekeeping of identities.
But you're not. You're American, with Irish heritage. You were born in America to American parents. You are super welcome to learn about Irish culture, about your heritage. You are super welcome to visit Ireland, visit the place of your fore-mothers and other ancestors. You can enjoy Irish culture as much as you want. Learn riverdancing and blast Michael Flatley all day long. You can even enjoy the bastardised commercialised version that is the totally fake US retail holiday "St Patrick's day". Wear some tacky green beads, put on a green hat, drink fifteen pints of Guiness! Sláinte! Have fun!
The one thing we're specifically asking you not to do, is to call yourself Irish. That's the only thing we're gatekeeping. You're Irish-American. You have Irish heritage. You have Irish ancestors. You have Irish family heirlooms. But you're not Irish.
Don't worry long ago I had my naivety removed by folks like you and no longer feel any fondness or interest for ireland or irishness, and passed none of it down to my kids. Ireland has never been brought up for a vacation destination where as I convinced my mom to gift me a trip to all of the UK upon graduation. Hopefully you will be relieved of the burden of having to deal with Americans with feelings of common bonds (like I used to have) after my generation passes.
I'm sorry your people have had to endure this wanting to connect from Americans and them trying to figure out if the weird/quirky things their family did come from your culture.