Modern equipment usually has a power controller chip that sits between the power button and the actual "power on" signal that's sent to the power supply. There

Modern equipment usually has a power controller chip that sits between the power button and the actual "power on" signal that's sent to the power supply. There's no direct physical connection between the button and the PSU anymore. The one I usually work with (LTC2954) does have an RC circuit attached to it to determine how long to hold the button to force a shutdown. And I'd wager a guess that this is the capacitor you're discharging by kicking the controller a few dozen times. But there's no possible way I can see that you're discharging every cap in the system.