Hello Brian,
Yes, precharging the capacitor would make for an orderly startup.
Supercapacitors are a fun technology. This morning I wrote an article presenting an example that closely matches what I understand your requirement to be - see link below.
Yes, cell balancing is a challenge that has received considerable attention. Please see these evaluation boards as they are generally applicable to both battery and supercapacitor technologies. Assuming you are designing you own PCB, the footprint is relatively small especially if you use cylindrical packing for the cells.
RE Spiral out of control: no, I wouldn’t describe it that way. Instead, the performance of a series string is dominated by the weakest cell. If all cells have nominal 2000 mAh rating but once cell it at 1500 mAh, you now have a 1500 mAh string. Also, voltage hogging can lead to cell destruction.
A real common real-world application is RC cars, airplanes, and drones. The series connected battery packs all have cell equalizing capabilities. This is typically a second smaller connection on the battery pack.
Looking forward to your pictures.
Have a good weekend,
Aaron
P.S. There is an argument for simplicity. Two lead acid batteries on trickly charge and two steering diodes would solve all of your problems provided you can handle the size, weight, and routine maintenance.