There are two types of technologies that are used for missile defense, hit-to-kill and those that use an explosive warhead. Each has had good days a bad, to put it mildly.
The Patriot missile system used in the 1991 Gulf War included an explosive warhead, and despite early press claiming how successful it was against Scud missiles (with hundreds of miles of range), it actually wasn’t. It probably hit several incoming warheads, while other incoming warheads simply disintegrated due to poor design.
On the other hand, the Iron Dome missile system, now in use by Israel to defeat small rockets with ranges of dozens of kilometers, seems to have done quite well with its explosive warhead.
The newest Patriot missile, which has been fielded for oh, call it ten years or so now (not sure off the top of my head) is a hit-to-kill missile. It works both on incoming missiles and against aircraft (though not quite as well against aircraft).
For intercontinental ballistic missiles, the US has had decidedly poor test results. However, even if the kill vehicle on the currently-deployed system were changed to have a warhead, that would NOT have any advantage. The main challenge with the current system is that it has a lot of moving pieces: the sensors must be quite accurate, the anti-missile missile has a huge amount of energy, the kill vehicle must maneuver quickly and efficiently, and it is possible for an adversary to throw up chaff and debris to try to confuse the system. Not to be too optimistic about it, but the least of the problems of the current system happens in the last second of the intercept. It’s everything that happens in the ten minutes before that is the challenge.
For what it is worth, the last couple tests on the US’s national missile defense system have been promising, with the caveat that the whole program has moved back to a “crawl-walk-run” strategy, instead of trying (and failing) to achieve realistic scenarios that were the basis of tests for quite some time. There’s a big realistic test scheduled for late this year, the first realistic test in quite some time.
And I can’t go without saying that in the 1970s, both the US and the Soviets were permitted by treaty to set up two anti-missile sites in each of their countries. The basis of these missiles was to detonate a nuclear warhead in space to destroy missiles headed for Washington or Moscow. The US scrapped this program, but it remained operational in the USSR/Russia for a surprising amount of time.