I took a closer look at the 74C04 Hex Inverter (IC103), which implements several NOT-gates. In particular the one that is needed by IC106 to sense that a plug is present in the 'both' inputs jack (Pin 11 = input and Pin 10 = output). So when a plug is present, this grounds the contact on Pin 11 and I verified that this is the case, since there is 0V on Pin 11, which means "high" or "1" in logic terms for the Hex Inverter. If this part of the Hex Inverter would work correctly as a NOT-gate, there should be around -15V on Pin 10, right? But this is not the case, it is also 0V. In fact it is 0V for every possible input jack combination, but it only matters if you use the 'both' inputs jack alone, since only then you need a logical 0 at Pin 10. If you look at the logic of IC106, this explains the behavior that I explained in my first post, and suggests that IC106 works fine.
So my best guess is a defective IC103 that does not invert the input at Pin 11, if it is a logical 1, so that Pin 10 is always logical 1. What do you guys think?
Here is a schematic, where I marked the NOT-gate in question:
https://ibb.co/fr4XZn