OK

You now want to just run a jumper wire from the blu at motor to a good ground w/key ON.
That should start fan on High speed. This is just a regulator by-pass test. If NO , then spin the fan by hand as you have this jumper wire hooked up...if YES w/hand spin, then you have some brushes hanging up due to wear/dirt . etc ...
Sometimes you can clean them up.......

If fan comes ON w/jumper , reg is suspect...but before condemning regulator, peal back the Yl and take a V reading down there at reg. just to make sure you don't have an open wire from /between the 3 wire plug to the regulator.

As you can see , the regulator controls the NEGATIVE side of the circuit [ a SWITCHED GROUND circuit], so the jumper takes the reg out of the circuit for testing purposes..but don't forget , NO trigger V at yl of regulator and you will get NO FAN b/c the reg is not being triggered....it needs that variable V to regulate the ground side of the motor.

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A Dalton