The air pump can only promote combustion in the exhaust stream when the mixture is richer than stoich, otherwise there's no fuel for that additional oxygen to burn. When you're stoich or rich, there is no oxygen left after combustion in the engine to promote combustion at the cat. Newer cars that did come with pumps came with electric pumps controlled by the PCM. In the case of my Corvette, it'd only run the pump for around 90 seconds or so. In that time, it also pulled a bunch of timing in certain areas to drive up EGT. Older cars used to use 2 way catalysts. Those ran slightly rich to keep NOx away, and the air pump ran all the time because of that to keep the cats going so they could get rid of HCs. So the only thing that'll happen in your case is that the cat will take longer to warm up if you ditch the pump. Warmup is the dirtiest time an engine runs, so that's why GM wanted the cats lit as quick as possible. Also why they go into closed loop so fast.