Thanks AndrewWynn! The issue indeed turned out to be related to the FSU.

I did the test shorting pins 5->2 and 4->1 on the FSU connector and the blower came on right away.

I then tried out connecting the FSU and it worked just fine! The blower came on... So, not sure if the FSU needed to be disconnected for a bit (it looks like it is constantly fed +12v) or if it was the FSU connector not being seated properly or having some rust.

So, I put it back together and it is all good for now. If it happens again I likely will just replace the FSU.

Thank you all so much for all your help. You guys are awesome!

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2001 BMW X5 3.0i, Automatic, Gasoline