1. Check the wire from the PCB to the ground point at the back side of the enclosure. I'd also try to unplug it from the PCB and listen what will happen.
I checked the wire going from pcb earth to ground on chassis, it looked fine. Also tried unplugging the ground connector on the pcb side, the hum became louder. I also tried sanding off the black paint from the chassis where the ground is connector to expose bare metal, this did not help.
Well... At least you are sure the connection is working.
3. To eliminate the possibility to induce the noise from the following gear - unplug the following gear from the mains (with Clarinet on) and listen what will happen.
The only thing I had plugged in were the shorted rca connector I had previously made, also tried with nothing plugged in.
Saying "the following gear" I was thinking about power amplifier!

... the bad or dry electrolytical capacitors, but then there should be no noise after powering off or unplugging from mains...
Any way I can test the capacitors?
Do not be in haste. There are not enough evidence for the diagnosis. I'd take serious attention to the Jim's words about shielding. It was the next thing I was going to suggest after checking the power amplifier influence. At this point of experimenting there are only 3 possible source of noise left:
1. Some leakage from the power amplifier (but you tried the device with other power amplifiers with the same effect).
2. Weak tubes (SRPP schematic used in the Clarinet is more demanding to the top tube in terms of cathode-heater voltage, so some good NOS or old tubes may behave not very good at this point). (but turning the device off should eliminate low-frequency hum almost instantly)
3. External EMI stray pick up.
IMHO, the third point is much closer in explaining the device behavior than others.
P.S. By the way - is the hum depends on volume control/balance control position?