The problem is that the general perspective of the UN by the US Citizenry is quite bad, largely because our politicians and news media began dissing and discounting the UN when they started saying things we didn't like.
It's more complicated than that--but there is actually such a thing as statecraft, and a lot of it goes on behind the scenes, and our news broadcasters usually don't care about it because it's too complicated to explore in a sound byte. Some of it goes on at the UN. Some of it makes a difference in developing nations, in the development of free markets, the somewhat successful (though we still have a long way to go) progress in the fight against slavery, work on preventing climate change, and even on occasion successfully prosecuting (or even just stopping) war crimes and certain malevolent acts.
It's a terribly flawed system, and I don't pretend to be an expert--but from what I've seen and what I know from people who actually know more about the UN than what they get on the nightly news, while there IS a lot of counterproductive or ridiculous activity, there is also a lot of good work that they do and that they try to do in various areas.
Those problems, by the way, happen here. Tens of thousands of slaves are brought into the United States every year. Climate change will kill us if we don't do more about it, opening up new markets (and stabilizing regions enough so that business can be done there successfully) helps us as investors, consumers, and producers, and a stable world is better for our Citizens abroad and at home.
So the UN does help us, in those ways and others. Sometimes it is ridiculous and is rightly mocked (a state of governance hardly unique to the UN), but more of the ignore-the-UN sentiment in the US comes from them doing things we don't like. We had disagrements with them on a few things, so our leaders and the press moved away from them, and now we're twenty years later. (I actually don't recall how many years later--I'm thinking, in particular, of the time we were indicted in the World Court for terrorism. We were backing an anti-communist regime with some CIA help, IIRC, back during the Cold War when we cared more about Communism than about Terrorism. The help included recommending they use terrorism to achieve their political objectives.) There have certainly been other things we've disagreed on over the years.
I'll also agree on the Veto point below--that the UN, and not another international body, is such a major international voice is very useful to us because we have the veto power in the security council. For one thing, other countries do care about the UN, and it's therefore useful to us to be able to have at least some influence over the UN's policies and positions.