The accents used to bother me, especially when the DCs first went to school and really picked up the local one (Chicago, so dreadful). Some words in particular made me cringe. They settled back into our own family accent (exH from another part of the US so didn't have the local accent) after a year or so.
If you heard them speak you would say my DCs are 'American', and you would be right. They were all born here. But 'home' for them is a certain little spot in the midwest, not Alabama or NYC or the Pacific northwest, any more than those places are home to me.
You are right to stop people from correcting them when they speak, and I hope you will be able to stop yourself from hating the accent too. Where they are living, the friends they spend time with, the jokes and fun they have, the language they speak, are their childhood and what their memories will consist of. Don't make them feel they are not really a part of it and that they belong elsewhere, or feel guilty in some way that they are hurting you by engaging fully in their own lives.
Don't mix up their 'home' with 'America' or whatever 'America' means to you. My DCs' memories are of T-ball, baseball, softball, baseball games in an empty lot, basketball - a 56-2 winning season for one of them, volleyball, schools, teachers, school trips, a game called assassins in high school, park district classes, going to the library, the ice rink, the ice show, the local pools and playgrounds, taking the El by themselves for the first time with friends, days on the beach, days at the Art Institute, vacations - yes 'vacations' - in Michigan and Wisconsin, Thanksgiving and Christmas, ice cream at the little local place, local pizza, all the friendly and hospitable neighbours and the parents of their friends, block parties - I do not want to throw a wet blanket on any of that. Their childhood would have been different in Dublin but what they had is the only one they will ever have. They will not be more or less my children because they were familiar with pumpkin pie from an early age and do not like plum pudding. My job is to let them know we are all in this together.