The aux coolant pump has nothing to do with coolant temp/engine overheating. It's a 'booster' pump for the heater. I'm not sure exactly how it's used i.e. whether it's simply on or off and switched by the heater or a variable speed/pulsed deal according to heater settings etc.
http://www.w124performance.com/service/w124CD1/Program/Climate/83-630.pdf doesn't exactly go into much detail but suggests to me (possibly wrongly) that it'll only run flat out when the heater is 'working hard' hence setting to defrost and feeling the pump?
If the aux pump isn't working i'd expect the heater output to be poor when the engine is idling and/or heat demand is high due to temp setting. Especially if the heater fan is set highish for demisting/defrosting. It'll be listed on the fusebox cheat sheet (both in your owners handbook and in the lid of the fusebox) somewhere, it's supplied by fuse #7 on mine along with duo valves and various other aux fans, HVAC bits and pieces and the reversing lamp
If the engine is running hot then you need to look elsewhere... tired/knackered radiator and/or main waterpump, dead viscous fan, dodgy thermostat etc. A better description of how the car overheats may help us narrow things down