"...In order for the fuel pump relay to activate, it has to have an ignition signal..."

One caveat - you should be able to hear the momentary pump charge when the ignition is turned on (before starter cranking). That initial fuel pressure priming is not reliant upon reading ignition pulses as the engine isn't even turning over yet. However, once you start to crank the engine, it's true that there must be ignition pulses present for the pump to start pumping again. So my point with all this is that if you don't hear that initial pump prime when the key is turned to position I, then a faulty ignition is less likely to be the cuprit.