First check from cluster hidden menu (or with diagnostics) if the engine temp is ever coming to normal 88-92. The gauge needle is on 12 o'clock 75 C->. For M57 engines the original BMW part for main stat (Made in Austria). All after market variants are poor quality made in Italy stuff. In UK you have Euro 3 model without EGR cooler so no need to worry about EGR thermostat. On automatics there is still another thermostat for oil heat exchanger but those are quite reliable and rarely fail. E53 has so huge radiator that I've removed the viscous fan permanently and not causing any problems if the front electric pusher fan works normally.

When diagnosing possible Webasto problems INPA has good live data to show what's happening or not happening.

__________________
E39 530dA -02 M-Sport Messing metallic
E53 X5 3.0dA -06 Sport Stratus grey
E70 X5 40d -12 M-Sport Space grey