Fine, have them choose one country or the other. Dual citizenship sounds really dumb assed to me. * I remember when my gramps quit going to the Italian-American club(taking me occasionally) "fuck them, all they want to be is Italian. I'm an American!" And so we were.
Dual citizenship kinda makes some sense, at least temporarily, for military members and others who marry and start families (or start families and then marry? Or fuck it don't get married but have kids?) with people from overseas. I don't know about abolishing it but maybe put a limit on it? Like when all your paperwork is rubber stamped and all your requirements are met to become a full-fledged citizen of one country or the other you have to pick one?
As for the second part of your post, that was the attitude of a lot of immigrants back then, including mine too. They came here to make a better life and BE Americans. Not to turn it into to they place they just left because they figured it sucked bad enough to pull up roots and strike out into the unknown. Ours anyway (Germans) kept our culture in the form of food, music, dance, and definitely stubbornness, but they were all about becoming this thing called American, whatever that is. They maybe didn't really know what that is but they knew what it is not.