the capacitor wont make any difference, the module itself is what controls the fan speed, the ECU sends a 3V PWM signal to it to variate the fan speed. The highest speed you get on direct running wont be achieved by the module as it limits the rpm to 90% of capacity so the fan doesnt burn itself out.
Almost all scrapped modules will be bad though - I have never found one that worked. Even outside pk. You can wire up a switch to it to disconnect the module so the fans run full speed once the A/C starts.
you also need to make sure the water pump is good, When I swapped one (4G63), I found that the impeller in the pump was cracked, the impeller would loosen on the shaft as the coolant warmed up - resulting in a hot engine.
test the fan motors first with an ammeter, do this on a full charged battery, you should not be seeing more than 11 amps of draw at full speed. If you do see more amp draw - you will kill even a new fan module. The remedy is to replace the fan motors with new ones along with a new module.
The fans should switch off when the key is switched off, if they remain working for any amount of time including 5 seconds, the module is bad.