Final update:
The cause of this issue has been identified. Despite what the previous Chase specialist told me yesterday morning about the failure being system-wide and requiring some time to resolve, I received a call from another Chase employee about it later in the day. This time the Chase representative told me the problem was likely related to the size and shape of the check and/or the positioning of its signature and account / routing number blocks, and also possibly the background color or pattern.
And that was in fact the problem. I subsequently attempted to deposit a fairly regular looking personal check using the QuickDeposit function and it worked fine. Maybe this will help someone experiencing the same issue.