Corrosion may still have a role to play in this. A voltage may still measure 12v with a meter but as the meter doesn't draw much current, it doesn't reveal the high resistance of a corroded connector. The high current draw of the motor however causes the applied voltage to drop below a level where the motor can run.

Have you checked the connector on the door pillar? Remove the battery connection (negative) before disconnecting the door connector, otherwise you may trigger an SRS fault.

P.S. did the odo come right again at some point?

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Wayne
2005 BMW X5 3.0d (b 02/05)
2001 BMW F650GS Dakar (b 06/01)