Power button behaving as OFF-(ON) momentary switch rather than latching- Troubleshooting
I suspect the circuit is not exactly like you have drawn (not surprising, due to the difficulty reverse-engineering the PCB). I suspect Q101 might be a PNP transistor to switch the supply to the regulator, and both the push switch and an MCU output signal can drive Q102 base, to pull Q101 base low and switch it on.


  how it might malfunction and exhibit these symptoms?


On some systems with an MCU (like your drum machine), the push-button only applies power as long as it is pressed. Assuming the MCU starts quickly e.g. 10s or 100s of ms, one of the first things the MCU does is to output a signal to keep the device powered on, even when you release the push button. (There are alternative techniques for when the MCU takes longer to boot, than the time you want the user to keep the button depressed.)

This "soft power switch" approach also allows the MCU to power itself off (e.g. if it detects battery low, or needs to save power, or any similar feature). You can probably see where I'm going:


No output from MCU to keep the power on = device powers off, as soon as the push button is released - exactly as you describe.


The problem is that any MCU-related malfunction can cause that type of behaviour, since the MCU must be operating normally for its code to run and set the signal to keep its own power switched on. Any hardware or software problem that causes the MCU not to run its code (e.g. failed firmware update, clock problem, reset problem - basically anything wrong) means that the MCU doesn't keep its own power switched on.

To test for this theory - does the device keep running correctly if you press and hold the "soft power" button? If no, then this hypothesis could certainly be true. If yes, then this hypothesis could still be true, but the range of possible faults are much smaller, if the MCU is basically running OK while you keep the button pressed. This might be pointing more towards a problem in the "soft power switch" circuitry instead.

A proper diagnosis (especially remote diagnosis) would probably need a schematic, although the test above (and using a scope to look for signs of MCU life, as well as more reverse engineering of the "soft power switch") might allow you to make some progress. IC102 on the bottom of the PCB might also be a regulator or other power device, and might therefore be worth investigating.