Is the power supply part of the amp board? My money was on the power supply, but if it's in fact part of the amp you swapped out, maybe not.
A switching power supply typically has "housekeeping" circuitry that runs off a winding of the main power transformer. But the supply has to be running for that to happen. There is usually no inefficient 60hz mains transformer, that would defeat the purpose. So in order to get the oscillator of the switch mode power supply (SMPS) started, there is a large value resistor that indeed charges a capacitor, often through a simple half wave diode circuit, to "bootstrap" the SMPS, that is fed directly from the mains. It usually holds enough charge to get the circuit started. But if it fails to start, it will just keep cycling, exactly as described. I've seen this behavior in many things powered by switching power supplies.
I'm really glad to hear you have help working on the sub, to avoid hurting yourself. Hopefully they are taking good care of you.