Thanks to a helpful hint from commenter Lynn H, our Kindle woes appear to have been resolved.
Short answer: it was the official Amazon case‘s fault.
Longer answer: we were not alone.
Even longer answer: We called up Amazon, explained the whole sordid story to the customer service rep on the other end, and then noted that theKindle had been behaving just fine ever since we removed it from its case. The CSR noted that might, possibly, maybe be the source cause, but wanted to go through the full trouble-shooting question-and-answer session to give the technical support guys all the information they might need, so we went along with it, and were promised that we would receive an email in a few days noting the claim’s disposition.
Two days later, Amazon emailed us a paid shipping slip for the case, a refund, and a credit towards the purchase of a new case – considering that we had to go through three separate Kindles, I guess that is fair enough.