Yes, because if those temps are an overheat condition, then it should have come on prior to that to control the temp. That fan can shift a hellava lot of air through the radiator.
Your car seems to be doing fine if the coolant level is OK and the temp has been OK ever since. I think you indicated the coolant was flushed in the last year(ish) anyway. Every 3-4 years on BMW coolant should be OK.
I don't know what would be a "trigger' temp for the fan, but it would normally be on before you noticed any abnormally high reading on the temp gauge.
114 is high but not excessive with aircon on under high load (low speed, high temp etc.) or stop/start conditions. Did the cluster temp gauge actually get into the red zone and light up the warning light?