re: "...why would a car that uses batch injection care at all about CPS?..."

The CPS isn't just for fuel pulses (which isn't synchronized to the crank anyway, as you pointed out). The CPS mainly is to replace a distributor sensor (Hall?) for ignition timing, viz. the crankshaft's angle. In these 2.4 (and 3.x california) cars, the processor controls the precise spark timing rather than the sensor of the distributor, which is there only to ensure that the cap-rotor is in alignment to conduct the spark's current. And the processor does all the subtle advance and retard adjustments electronically, rather than mechanical gizmoes (belows and weights) inside the distributor.