I would try to spin the blower by sending a jet of air down the dash vent. Blow mode of vacuum or maybe leaf blower. Do while measure volts on blower ±. It should generate some voltage. I do not think there's any Electronics from that connector to the blower. You may be experiencing a dead spot. If you get the fan to turn a little and get it to run, take a reading of AC volts on blower it should be almost zero. Even better if you have oscope. Open commutator or winding will create voltage spikes.

I just looked up to the top to see what forum. E53 is much harder than e70 to deal with blower motor. I think I'd have a portable vacuum in my car to jump start the blower for a while before I tore the dash apart to swap blower.

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2011 E70 • N55 (me)
2012 E70 • N63 (wife)