Right, the controller is a PWModulator...so, feed power from it varies in order to maintain fuel's low pressure within its line. The car's battery will supply 300 to 1000 amps(range for most car batteries)...so way more amps than what the pump normally receives when it was directly connected to the battery. I have a Torqbyte PM3 between my controller and lpfp and it takes the signal from the controller and increases the amps up to 36 amps for the upgraded lpfp.
So, Pudding is correct in that even a badly worn pump, if not completely shot, will run given more amps.