Ok, my report-back from the Apple Store Genius Bar:
Earlier, while I was waiting for my Genius, I'd worked in an app, then switched to some other apps and worked there for a while so the problem would be ready for him when he got here. Then he did arrive and I explained to him what was happening. He told me he had never heard of the problem -- disappointing, but maybe not surprising, right off the bat. Anyway, I then showed him my iPad and switched back to my original app that I'd had in suspension, and -- wouldn't you know it? -- it did NOT reset. Grrr...
But then I switched to another app again did some work there, came back -- and he did see it reset. He saw it, and was surprised. He said something about "well, maybe some apps need to reset because ...", blah blah blah. OK, whatever.
I showed him our discussions here and he did read quite a bit of our messages and I think he was convinced there was indeed a problem.
I told him about our theory of running low on memory, and he hooked up my iPad to a diagnosis machine and did find that my iPad was quite low on "application memory". He noticed that on my 16 GB machine according to Settings -> Usage I had only 1.3 GB free. He commented that that is extremely low and that they recommend always keeping at least 10% of disk space free. He suggested that could potentially be causing the problem: we all know that sometimes RAM needs to access and draw a bit from disk space in order to work correctly if it's low on actual RAM (well, I'm doing this from my knowledge of Windows but according to him the same principle holds).
He also noticed that I have 109 apps installed, which he said is extremely high (really?) and wondered if that could be contributing to the problem.
He suggested clearing out some disk space when I got home, and seeing if that takes care of the problem. So in the last day or so I've uninstalled some large apps and deleted some data for some apps, and I'm now pleased to report that Usage -> Storage now tells me that I have 2.8 GB available and 10.9 used. Definitely more than 10% free now. And ...[drum roll here].... I'm still having the problem. No difference at all.
Does having 109 apps steal memory also? I'm not sure why it would as they're not loaded.
Back to the drawing board.
Additional thoughts welcome.