These are good questions. Like all good questions, they don't have a short answer. I'll just touch on a few things, and maybe others will chime in.
Let's start with history. Ronald Reagan was convinced by Edward Teller that missile defense was possible. He felt that a working missile defense would make nuclear weapons obsolete. So he got a program going. One answer to the first question is that once going, programs have a life of their own and continue. That has to do with government funding generating its own lobbying organizations, namely those that are getting the funding.
Let's look at history from the Russian side too. During the Cold War, it was apparent that the development of anti-missile missiles could lead to even more competition, if one side developed an anti-missile missile that worked and then the other developed something to overcome that. There were more abstruse arguments about second-strike capability and its role in deterrence, but I would have to look that up to get it right, and I'm making this a short answer. This was a problem for both the United States and the Soviet Union. So they agreed on the Anti-Ballistic Missile Treaty, which said, in effect, that both sides would agree not to develop anti-missile systems. George Bush withdrew the United States from that treaty in 2002. Russia/Soviet Union has always been more sensitive about this issue than the US. Recent comments from Russia, however, probably have more to do with the election they held last weekend.
There is an element of bluster on both sides.
I'm up to Reagan. After that, all presidents except possibly 43 knew that all analysis shows that anti-missile defense is unlikely to work—the most optimistic scenario was for one portion where it was unlikely that we would hit the missile, and if so, it was unlikely we would damage it. Both the scientific establishment of the US and presumably Russia know that this is a tremendous waste of money. So why does it exist?
It's probably not entirely correct to focus on presidents; Congress votes on the funding. That said, it was Reagan who got the thing started, and presumably strong presidential action could help to end it. But it's Congress that needs to be convinced that missile defense is not worth funding; that could be for scientific or political reasons.
One reason why it exists is that defense contractors and the representatives and senators from their districts have become accustomed to the $10 billion or so a year that they get from the program. There is also the argument that eventually it can be made to work, and the spinning of test results in the most optimistic ways possible, which are, it is true, debunked by missile scientists not in the pay of the defense contractors.
If a president argues against something like this, or someone in Congress votes against it, that leaves open the possibility for an electoral opponent to argue that they are undermining the safety and defense of the United States.
Those are simple answers. As always, there is much more to the debate. Maybe I should write a post in House Blend.
If one is preparing a defense, it is a good thing to consider what one is defending against. That issue has been muddled in the current standoff about missile defense systems in central Europe. The Americans claim it is Iran; the Russians think otherwise. Another concern that supporters of missile defense list is electromagnetic pulse. This is the idea that a country would launch a missile with a specialized nuclear weapon that would explode at high altitude over the United States and knock out all electrical and electronic equipment with its electromagnetic pulse. This has been debunked, but Newt Gingrich mentioned it in Saturday night's Republican debate and has been vocal about this "danger" for some time. The Heritage Foundation has written a lot about this, the American Enterprise Institute somewhat less.
There are two kinds of missile defense: theater missile defense and ballistic missile defense. Theater missile defense is more local, against missiles like SCUDs, and it works reasonably well. It's the ballistic missile defense, the kind that would be needed against electromagnetic pulse. William Broad considers Gingrich's claims in the New York Times today. He makes another point: it would be necessary for an adversary to have the missiles that would need to be defended against, and Iran and North Korea don't have them.