If they are supposed to be the government they need something to govern.
The only remnant of the USA is the Enclave itself. They are a government (and a fascist one at that) without a country to govern and thus without a proper purpose to exist.
The USA have been destroyed, history has moved on. It's time for new countries to rise (like for example NCR).
They are going to govern what they build out of the ashes of the War. In order to do so, they need a stable foundation, which they can't ensure unless everything impure is dead. You horrify me by your comments, about America being dead. It may be so, but that doesn't mean it can't use a reboot. America, up until the War was
the best place on Earth, it deserves to be rebuilt, and if what it takes to do so is such measures, I don't blame them, instead I stand by them.
Most of you are frightened by the fact that people are being killed because they're irradiated. Would you step into an elevator with another person knowing he was irradiated? I seriously doubt it, and if you would, then I doubt you fully understand the effects of radiation. If these were better days, they would have gone through therapy of sorts, but unfortunately there isn't. The greater good calls for further destruction.
To put it into perspective, if a 10 floor building is destroyed and the first floor is still standing, but crumbled, do they start building on that floor? Or do they clean up the rest and start from scratch? Answer that yourself. Basically, you're all asking for us to stick with what's going on and accept it, if we do that, we will be stuck in a Wasteland forever.