This sounds very similar to what I’ve experienced on my 98 2.8.
First time:
Viscous fan was weak - cover overheated at idle but now hem moving.
Second and third time:
Water pump impeller loose on the shaft - so spun weakly at idle but fine when driving /air flowing.
This happened on a replacement metal type pump so they not infallible. Twice.
Now like yours, my car has an aircon and the fan did work when the AC was on BUT it was wired up incorrectly from the factory meaning it didn’t kick in when water temp rose (like it should, as these car do work fine without the viscous and just the aircon fan - that’s how mine has been for years now - not on track, another matter though).
Basically the aircon fan activates if there is insufficient cooling from the viscous (like if it fails or sat idling in very hot conditions)
I’ll dig up the link to the aircon fan issue for you, easy to do once you have the pics.
You of course have another issue be it water pump and or thermostat. These really are service items on BMW so I would get both changed. Also do the thermostat housing (they do crack I know!).
Not expensive and of course it’s off when the thermostat is being changed.
I should have it all done at once would have saved me multiple trips the garage and worries about breakdowns (garage weren’t BMW guys so didn’t think to tell me to do it all at once, hindsight is a wonderful thing!).
