The little boards are the Voltage regulators
You are in the US so using 110V
Look at the wattage V * I roughly [ It is not really true because there is a difference between AC and DC ] you use at home.
12V * 250A = 3kW
divide through 110V = 27A
Here in EU we use 230V which gives 13 Amps, these very short 3 cm 1,5mm2 cables are used everywhere for AC till 16A.
And we are talking about just seconds.
Only at charging the capacitors for the first time only the caps get warm the cables not.
For the connections between the caps and for the longer cables I used 6mm2