Minor update: after I replied to Capital One's canned email with a few choice words and requested escalation to a supervisor, I received a slightly more meaningful reply. The customer service rep said that my complaint "was forwarded to the appropriate department for review", and offered a $30 credit as a goodwill gesture for the delay and inconvenience. Still, they said that I need to wait for the claims processor to make a decision. While I certainly appreciate the goodwill gesture, it doesn't solve the basic problem: I now have no confidence in the insurance protection offered by Capital One.
As a result of this, I am already putting all the purchases that may require insurance through AMEX Rewards Gold. Since with that card you get two points per dollar spent on groceries, gas, travel and drugstores, there's no difference between its yield and the Capital One Aspire for these purchases.
If you ask me, Capital One's choice of travel claims processor was a pretty dumb move in terms of cardholder retention.