I don’t think that’s an accurate description of the American people, let alone the American government. Yes there are some nut jobs making the news, but I’m sure there are plenty of British, French, German, etc. nut jobs. For example, when people in France riot and burn cities, overturn cars, etc. we Americans think “what the hell is wrong with the French that they can’t disagree without rioting?” We’re as wrong as your description of us is. And that’s not even getting into the real nut jobs like say, North Korea or the Taliban. As far as being trusted or responsible enough to have them, all we can say is that we have more than anyone else has and have had them longer than anyone else, and except for two bombs dropped during the mother of all wars when nukes were very new, we haven’t used them, even during very tense situations involving a similarly armed neighbor.
But ultimately the reason we get to have them and no one else does is that a whole lot of countries like us having them and count on us as a deterrent to China, Russia (yes, still), Pakistan, North Korea, or whoever else you can think of. They liked this arrangement so much that the vast majority of countries decided to sign a treaty basically saying that they would not seek nuclear weapons if we promised to punish the holy hell out of anyone who uses them on them and to help them with peaceful nuclear technology.