I'm familiar with the concept of nuclear winter. But again, try to find proof that it would destroy all life on Earth. You will find none. The Cretaceous-Tertiary ELE caused similar effects, and only ~50% of all species were destroyed. Most of them were highly specialized organisms with little adaptive capacity.
Also, I suggest you do some research on nuclear ordnance, and you will find that thermonuclear warheads (the kind of nuclear weaponry needed to cause the massive fires needed to trigger a nuclear winter) are relatively clean in comparison to their standard counterparts, so the radiation clouds would not be as bad as you presume.
Yes, it would be a catastrophe, and civilization would perhaps be set back a few hundred years, but mankind would not become extinct.
And how are you so sure that "vaults" are impractical? I don't have specific data on what the US govt. contingency plans for nuclear war are. Somehow, I doubt you do, either.
Right now, the cost of putting stuff in orbit is too high, and the benefits are not such a big deal. But give it some time, until going into orbit isn't much more expensive than going to Taiwan.
Also, granting exclusive rights only encourages monopolies, and that in turn tends to cause the system to stagnate. Let them rip each other apart for being the first to build a mall on Mars.