Typically it has to get nearly to red, which does not happen very often on
BMW.

Doesn't your car have a fan that is driven by the fan belt(s)? Mine does,
but I have the 3 Series not the 5.

If you have a fan that is driven by the belt(s), then your fan clutch is
toast. The clutch is a viscous coupling (VC) that grabs the fan when the
engine temps increase, and this causes the fan to pull air through the
radiator faster. When the car is moving, then air is pushed through the
radiator pretty efficiently because of themovement, but when the car is
stopped, then the fan is needed. If the VC is not working properly, then the
engine will heat up while the car is stopped, then the temp will drop again
as your speed increases, assuming you drive normally.

Seems quite common on older BM for the auxiliary or a/c fan to stop working.
On mine it's quite obvious that the motor has siezed. The fan is very stiff
to turn.
Another possible reason for it not coming on, is a burnt out fan resister.

One of the dedicated BMW websites, can't think of which one offhand,
explains how to remove and check the fan and resistor on a 5 series.
Mike.

On the E39, it's pretty common. The acid test is to switch on the AC - it
should then run continuously. Dunno about the E34.

Most commonly it's the motor itself that fails or seizes. Although it
should be easy enough to 'hot wire' it and bypass the sensors and relays,
etc.

I'm sure if you do a search of the various forums you'll find instructions
for changing it.