And, in addition to resources and the environment, we also need backup civilizations out of ICBM range so everything isn't wiped out when one of the shitheads in charge of the world hits the big red button
I find it hard to conceive of a nuclear war that would make Earth less habitable than Mars (or the Moon). Even if you take into account nuclear winter, massive fallout, large sections of geography glassed over, 99% extinction rate, it's not as if nuclear war would actually rip the atmosphere off the planet. There's a pretty big difference between "if you leave the bunker without protection, you might die of cancer in six months" to "if you leave the base without a space suit, you'll asphyxiate in 60 seconds".
No, I don't think we should go to Mars with any practical objectives in mind. Someone is going to need to make up some sort of plausible justification to generate the necessary political will, but I doubt the reasons will be legitimate.
I think we should go to Mars just for the lulz. Humanity needs to get out of the house once in a while, try new things. We can figure out if there were any tangible benefits afterward.