I vote for thermostat stuck open.
If the car still has the original plastic thermostat cover, I would put on a new thermostat cover while I was at it. I re-used one last year and ended up pulling off the fan again last month to put on a new aluminum thermostat cover because the plastic one leaked.
This is not terribly expensive, though it adds up if you are like me and tend to replace every plastic or rubber part you touch so you don't have to work on the car again soon. When I did my water pump, I replaced the thermostat, the radiator, expansion tank, and all the hoses. The only part I failed to replace was the stupid thermostat cover.
Water pump, Graf (if needed) $61
Thermostat, Wahler or Behr $20
Thermostat housing, aluminum $12 around $15 for plastic
Gallon of BMW antifreeze $20
If these parts are original, they are overdue for replacement anyway. Actually, if they have been replaced once and the car has over 200k miles in it, they are about due to be replaced again. If the expansion tank is original, It is crumbling inside by now. $40
Hardest part of the job is making the tool to hold the fan still so you can get it off. You need a long thin metal bar from the hardware store, a drill, and a grinder.
The fan is left-handed thread.
If you pull the radiator, and you leave the transmission cooler attached to the car like I do, unbolting the transmission cooler from the radiator is a bit of a PITA.