Update:

I have narrowed down the root cause to the shift register. However, it was not the way that I thought of before.

After further research I’ve learned that the shift register in question was supposed to remain in its last state when the rig enters the standby mode. It’s supposed to stay that way, even without the data or the clock signal, as there should be no strobe signal going in. This is the normal behavior for shift registers.

What seems to happen during the anomalous state is that a stray voltage spike goes into the strobe signal pin, opening it up for the data input, which it take as all zero, since there’s no data going in. This is why all its output pins have 0v instead of the expecter 5v.

I made the above discovery basically by replacing the shift register with a new one. The original CD4094B is hard to come by, and I don’t trust the Chinese clone, so I opted for a more modern replacement, the CD74HC4094. According to their datasheets, they both should be virtually interchangable. Turned out it indeed was the same, down to that damned loud static noise!

At some point, I accidentally shorted the strobe pin with the Vcc and the loud static noise appeared. I then removed the shift register and tested for connection between the two pins thru their empty solder pads. To my surprise, it only made connections with the logic unit connected, with resistance within megaohm magnitude. They make no connection at all when the logic unit is detached.