Thanks for that explanation, but I am still somewhat confused why that would be. In the case of mining multiple chains with PoC, there is not a "Nothing at Stake" problem that I can see.
Yes, I also agree with that. However, the reason why many think there is a Nothing-at-stake problem in PoC is probably the way how it's mined today. As there are only minor coins (Burst being the biggest but it's already a >top100 coin) existing now, not many resources are "burnt" to mine proof-of-capacity coins. So people can mine with their free HDD capacity, and thus the earned rewards are practically "free coins". In this situation, it is very cheap to mine on several chains and thus to attack the coin, cheaper than to mine a minor PoW altcoin. (One could call this attack "little at stake" instead of "nothing at stake").
This however changes when mining is carried out in an industrial fashion, as then the capacity to handle maintainance costs (HD failure, principally) become the factor which make a miner (and an attacker!) competitive in the mining game.
What I am trying to figure out is if there is any reason why resource usage in a non competitive environment such as PoC (or indeed an Proof of Useful Work algorithm) is any different than resource usage in a competitive environment, such as the existing SHA-256 PoW, that allocates rewards statistically rather than on a guaranteed basis. They seem to be identical to me from a mathematical point of view, but there is a group of people who insist they are not.
Why do you think PoC isn't a "competitive environment"? There is no guaranteed reward in current proof-of-capacity algorithms. The only coin I know that really had a "guaranteed reward" (but for transaction processing) was Timekoin, which mainly is defunct because it was too inflationary to gain any value (and also had other problems, as the double spending problem was not really solved).
I haven't thought much about the resource usage difference between guaranteed-reward and "competitive" concepts, because "guaranteed-reward" concepts for coins are problematic from other points of view, mainly because of the difficulty to prove that you really participate, and the complexity/scalability once the full node number increases.