Thank you for your thorough insights, especially about the ICS issues.

Fortunately, I've worked in IT long enough to know that new releases and OS "updates" always have defects; therefore, I refused to update any of my software since my phone was perfectly stable.
Also, I confirmed it's not a battery issue-- I tried a known-good battery and the old battery powers on my new-in-box S2 (bought as a backup for when I was going to root the one that died).
I hadn't formatted any system partition and unfortunately, didn't get to root it for backup purposes. It was still running AT&T's stock Gingerbread.
Ultimately, I suspect the issue was the 100 degree weather to which the phone was subject in my cargo pocket.
So on the assumption the phone overheated, I'm looking at either sending the phone to a forensic recovery center or to a repair center that "may" be able to diagnose the defect and repair the phone without data loss. Of course, I'm having a hard time finding a reputable, competent company where my life's data is protected.
(And unofficially, I've already had the phone apart to look for visible damage to the board and components, but I didn't see anything similar to the fried motherboards and chips I see in the PCs and Macs that I diagnose as part of my profession...)
So I know the data is there, I just can't get it unless I send it somewhere that does have chip-reading capabilities... or I risk soldering the old chip on the new phone's board by myself, not something I'm willing to try right now.
I'm hoping to find some "crazy smart" electrical engineer/genius/savant who can diagnose the problem, recover my data, and repair my phone instead of having to resort to only forensic data recovery...
There has to be someone on the internet that has successfully resurrected a smartphone with a hardware "transplant". I just have to find that person and compensate them for their expertise...