Hi smoliarm,
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I think C1 Chondrite is in fact the term for the meteorite fragments that we find that I guess have gone through extreme heating on arrival.
Fortunately for us, this is not true. Only the fusion crust of meteorite experience some heating (some hundreds °C, depending on entry mode), and this is VERY thin (~0.5 mm) outer layer. The rest of meteorite does not see any heating above 50°C at all. It's hard to believe, I guess, but this is a fact.
I processed and analyzed a number of meteorites, both stones and irons. They all contain a lot of troilite - mineral which decomposes very rapidly even at 200 °C. And this is not the only evidence that meteorites do not go through bulk heating during atmospheric entry.
I understand, that this is something contra-intuitive, given the violent-looking meteorite entry, but this is a well-established fact: meteorite fragments reaching the ground did not see any serious heating, just a brief mechanical shock.
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My (weak) understanding is they are hard to find because the asteroids they come from tend to explode when they hit our atmosphere, due to the volatiles they contain.
No, it is because they are very rare. And, once again, there are strong evidences suggesting that CI chondrites do not originate from asteroids. Therefore, it's not a good idea to use their composition to estimate result of asteroid capture mission.
BTW, the explosions here are NOT due to "volatiles" in common sense. It is a release of huge kinetic energy, which results in rapid heating of *unlucky* part of meteor up to 3000-3500 K. At such T, for example, metallic iron IS volatile just as water ice at 400°C, and of course it explodes shuttering the meteor body apart. But the rest of material (the *lucky* parts) do not see heating at all.
What I am really interested in is the amount of water available in asteroid material before it becomes a meteorite.
Trust me, the amount of water in parental asteroid and in meteorite originating from this asteroid is essentially the same.
Here is a quote from that Keck paper that initially suggested capturing an asteroid:
A 500-t, carbonaceous C-type asteroid may contain up to 200 t of volatiles (~100 t water and ~100 t carbon-rich compounds), 90 t of metals (approximately 83 t of iron, 6 t of nickel, and 1 t of cobalt), and 200 t of silicate residue)
It is just an unfortunate coincidence of letter "C" in C-type asteroids and in C-chondrites, they have nothing to do with each other. Especially, this is true about composition.
It is a common misconception that C-type asteroids are made of C-chondrite material - this is NOT true.
In fact, we do not know for sure that C-type asteroids are carbonaceous at all. The spectral data alone do not allow to make such conclusion, and we do not have any other data so far.
I do not know, how the authors got the above estimate of asteroid composition, but I can assure you it does not follow from meteorite data. All I can tell you -- they would have pretty hard time with reviewers and editors if they try to publish this in journals like EPSL or GCA.
One more detail - most of "carbon-rich compounds" in carbonaceous chondrites are carbides, and they are REFRACTORY, not volatile.
I admit it is probably risky to base my understanding on a single paper whose focus is not so much on space science as why we should give them 2.5 billion dollars 
Well, in my opinion asteroid capture is a great idea which has huge scientific value. So, there is no risk from scientific side. With asteroid mining it is quite different story. But, even if they fail the commercial part of the enterprise, we scientists still get our samples, no risk

So, what the hell, give them these $B and let them fail water extraction