Ok, here's something for all of us with this issue to consider/discard -
I'd been living with my low battery for a month, when it went completely dead. Charged it last Friday night. Turned it over Sat pm, started up first time, no problem.
Wife drove the car Tuesday, then it sat until I got home Thursday. I turn the key, and it sputters for a few seconds, before firing up. I think "WTF? I just charged this dmn battery!". Then, the stereo fires up, so I know that the wife left it on to power-down when she took the key out of the ignition and opened the door (cos, why wouldn't you, right?). SO, remembering this "Shaker Amp" theory, I decide to do nothing, other than turn the stereo off, as I pull up into the garage, once I was done driving for the night.
Now, normally, with the battery this depleted, I'd get the same start issues the next morning. Well, I went out at 11 this morning and, wouldn't you know it, the thing fired up STRAIGHT AWAY. Having lived with the low battery for weeks before I was forced to charge it, I know that this is NOT what was happening before.
There might really be something to this amp theory.