Ignition switch can also cause this. You can measure from the EWS module connector if starter activation voltage is coming from ignition switch. If yes, you could short the pin 1+2 thickest wires on EWS module to rule out its role. The module internal relay can fail even if the key recognition works correctly. If the key recognition does not work this shorting should activate the starter if starter motor itself is functioning but won't allow running if EWS does not give DME allowance for fuel injection.

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