Why does the United States continue to host and participate in the United Nations?

The UN has outlived its usefulness. Once it was a showcase for participatory government. Today, it is populated by thugs from all over the world looking to embarrass the US and get whatever freebies they can. We should stop funding and send them all home.

Well the UN exists because it is the only game in town.  It should only replaced not removed.  Just disbanding it without an functional replacement is a very bad idea.  Ideally reforms to the existing UN are the preferred avenue.  A functional replacement with a large number of participating nations would also be acceptable.  The terms of membership should be lenient because it is about the mission of inclusion and moving forward.

Because NOT having the United Nations is worse.  The UN is a useful pressure release valve for countries to vent their frustration at one another in public without using guns.  It is good for the world to have a UN, but if we pull out, we will be picking up bits and pieces all over the world as our interests are threatened continuously by countries that would rather fight than communicate. Why the US?  Because we have worldwide interests.  We are vested in the stability of the world.  That's a fairly unique position. I would love to see reforms to the security council, but realize that all of the times we have been frustrated by a veto from Russia or China, they have been frustrated by a veto by the USA (usually in a resolution against Israel).

The easy answer is that the US hosts the UN because NYC and the US generates about $1B annually from its presence -- it's just good business to have the UN in NYC. And as NYC crumbles, it needs all the help it can get.   The other reasons, as outlined in the book "Diplomacy" is that the UN provides lots of cover for the US, allowing it to do many things it  might not otherwise be capable of completing. For example, countries that hate the US and would never allow the US on their soil might be more amenable to the UN, even if the contingency were mostly US soldiers/scientists/etc. The UN provides a good fig leaf for the US to get its work done, both good and dirty. In the book "Diplomacy", the State Dept estimates the UN is the equivalent of the US having to deploy an entire army in various places worldwide, with the associated expenditure and risk, without the US being associated.   So the value to the US is huge. In many ways it is a crippled and foolish organization and mostly irrelevant except when it's needed to present that fig leaf -- how many times did Bush scream about Iraq ignoring a single UN sanction and yet he never said a word about the 13 or so similar/equivalent sanctions Israel has received. The UN could be worthy and could accomplish more but the member states in microcosm show all the greed, pettiness, meanness and vile of their full size counterparts.

The answer is in your question: "China and Russia have veto so nothing gets done anyway." This is a (very loose) model game for President You's behavior when a foreign policy crisis starts. In this game, you first get to respond to the crisis by making a hawk/belligerent commitment or by making a dove/pacifist commitment. World events then either spiral toward war or settle toward peace. If things are headed toward war, you get one last chance to either renege your promise or honor your commitment. I've colored the costs and benefits in the bottom area to reflect a specific (but relatively common) confluence of factors: The public favors a hawkish response, The public doesn't like leaders who don't honor their commitments, and You (as the President) don't actually want to go to war. You can see that even though you don't want to go to war, you don't want to sound like a dove at first - you end up committed to the side of the tree with only bad outcomes. If you take a hawkish line, there's still a chance you can get an outcome you want - like the one on the far right of the tree. But you can't be sure that world events will cooperate, and they might force you into the Hawk-War branch, in which you might be stuck either going to war or backing out on your commitment. This is where the United Nations comes in handy. The United Nations gives you an additional choice in the Hawk-War branch in which you can choose to mire the intervention process in UN bureaucracy. Then you get to continue to sound like you want war without having to actually intervene. In fact, the United Nations guarantees that you will never be stuck starting a war you don't personally actually want to have. One implication and two disclaimers: Implication: The President will always choose to sound like a hawk if the public wants him to. He is now able to guarantee that he will not have to go to war no matter what public promises he has to make. This makes it impossible to tell from the outside whether the President actually wants conflict or not based on his language. Disclaimer: The result I just showed only holds when the President does not want conflict. In this game, a President who wants to have a conflict can force intervention on his own. Disclaimer: I don't study international relations and I don't know this scientific literature at all. I just slammed this together in the morning before class. It represents a very slight formalization of my opinions as a bystander, not a summation of the work of people who actually study this.