With big fan picture (high fan speed) pushed does it turn on when doing all your tests as the regulator will default full 12v+ to the blower motor. Just like when defrost is set position will default to max power.

In auto if the sampling motor behind passenger dash and the one under the wiper under the hood are referenced to tell regulator to change voltage to the motor.

There is no relay for the system. If it has to be reset there is something wrong with the CCU (climate control unit/pushbutton assembly) or that regulator is toast if it cant get voltage to the motor if the 3 pins at the tower are good voltage wise. I am pretty sure id bet its the regulator if its not bosch.

I feel youre pain. Good luck.

If all else fails change that fuse you just never know.