When one of the plates of an isolated capacitor is grounded, does the charge become zero on that plate or just the charge on the outer surface become zero? The charge on that plate becomes the same as the charge on Earth.
Connecting one end of your capacitor bank to ground (through a resistor or not) isn't going to discharge the caps. You need to connect the resistor across the caps for that to happen. To elaborate on @brhans comment, with the earth switch the circuit is fully isolated and floating with respect to ground.
The purpose of a capacitor to ground is preventing your device injecting noise into the mains. Switching power supplies are notoriously noisy and they have to add these filters to pass EMI.
The better way to test is to use the resistance ranges of your multi-meter and check for very low resistance, say less than 1 across the capacitor. If your multi-meter has a null or zero function use this to zero out the lead resistance to get a more accurate indication of very low resistances.
You cannot charge a capacitor to a higher voltage than it is being supplied with. If the voltage present across your bridge rectifier were 2.5V, the capacitor bank will charge to 2.5V, and no higher. The "smoothing supercapacitors" present the same problem as the capacitor bank in the first circuit. They will act as an open circuit once charged.
If your console runs OK it is very unlikely that there are shorted capacitors on the supply lines. Usually you would also measure the supply voltages to verify the circuit. A shorted capacitor on a supply line would drop the supply voltage to very close to zero and disable the console completely. Re: Capacitors shorted? Is this normal?
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