The fan and switch are the only active components in that circuit, so I would test their functionality with new wire. When I replaced my non working fan, I found the 3500 RPM operating speed a little noisy so I added a 12v PWM speed controller to slow it down. That's a possible hidden feature that test wiring would eliminate.
I do leave my fan switch on 365 and let the thermostat control the fan operation. I guess that qualifies as AUTO mode.
Mossey