Ok, received two new servos today and did some testing.
First: For the record, my original servo was a Kookye DS04-NFC 360 degree continuous servo.
New Servo #1: Feetech FS5106R 360 degree continuous servo
-Testing result: Does exact same thing as Kookye, i.e. it makes sudden jump of roughly 20 degrees the first time power is connected. The only difference is it goes clockwise instead of counterclockwise.
New Servo #2: Longruner MG (metal gear) 996R regular servo (not 360 degree)
-Testing result: Also jumps counterclockwise about 20 degrees as soon as initially connected to power, BUT it hits its own internal stop if it happens to be close to that point when powered up. If so, it doesn't move any further, so no stripping of gears. Of course, this is basically the definition of a regular servo, i.e. it won't move past the range for which it is designed. But it does technically solve part of the problem (stripping gears). Secondly, it has metal gears, so probably not possible to strip them under normal conditions anyway.
It still does not solve the problem of the servo moving suddenly and frightening the unsuspecting user, but I can live with that (or rather, I have given up on that one since the transistor approach is too much for my technical skills at this point).
I guess you can conclude that I failed to solve both of my problems with my original 360 degree servo, but after making some adjustments to my mechanism, I can now use a regular servo and at least avoid the stripped gears problem of the 360 degree servo.