Preliminary research showed this:
It is actually a 3.3V circuit, and as far as I can see, the problem should indeed be with the 0.22F 5.5V supercapacitor. After opening it up:
The confirmation. Well, I cleaned it up and replaced the .22 supercapacitor:
For the record, the accident waiting to happen was a
Starcap, manufactured by Korchip, a subsidary of Samsung. The replacement is branded Eaton (which seems to be a rebranded Panasonic EECS0HD224H), according to whose website...
Eaton's KR family of supercapacitors replace button cell batteries (and the battery holder) for real-time clock backup power, and provide ridethrough for battery operated devices to allow time for battery replacements. The KR has excellent life time in typical 3.6 V systems. The family is RoHS compliant.
...it can be replaced with a CR2032 battery, which certainly would prevent this from happening again ~10 years from now.
But I digress. After replacing the supercapacitor it showed 3.3V at one end, yet the unit still wouldn't turn on. I'm going to replace the 4+3+3 resistors and the diode next, and see what happens.