I called Apple support this morning and was helped by a nice guy. He put me on hold while he discussed the problem with a senior technician, and when he came back he basically said I needed to go into a store. (As an aside, the senior tech had heard of this issue, so hopefully engineers will catch word soon too.)
He helpfully made a genius appointment for me tonight, from which I just returned. They basically handed me a new iPad (sans cables, power adapter, etc. but at least it wasn't refurbished) after I demonstrated the problem. It only took 30 seconds for a YouTube video to crash the system, which for the first time actually made me happy. I even got the colorful, animated corruption on screen.
I'm syncing the new iPad now. Assume all is well unless you hear from me later tonight. I recommend others with this problem follow suit.