[ferongr]

>Maybe the car has an AC compressor or a pump to cool the battery during charging, which draws directly from the charger when plugged in

No it doesn't. A/C compressors in EVs are 3 phase AC powered by an inverter supplied by the traction battery.

The contactors in an EVSE open and close at practically zero current, the OBC ramps the current up or down accordingly. And as another sibling comment noted, the OBCs are PFC corrected and act as a largely resistive load.