The sensor is a variable reluctance sensor, similar to most ABS wheel speed sensors and shaft speed sensors. The sensor does create an alternating current signal through induction, the voltage strength depends on wheel speed and air gap between the sensor tip and reluctor wheel (toothed ring). The signal strength drops quite rapidly if the sensor tip is outside the specified airgap.

I don't have the spec for the air gap, but it is usually somewhere between .015-.040 inch. You don't want it contacting the tone ring.

1/5 inch = .2, which is way too much. There must be some way to remove the bracket or get the right part that will place the sensor tip in the correct location.

The light may be on for a wiring fault, because it would have to be driven to realize an air gap fault that was causing a missing or weak signal. If the light comes on AFTER you start rolling, then this sensor could be the problem. If the light comes on as soon as you start up, before rolling, there is probably some type of wiring fault affecting a speed sensor wire somewhere. That will require some testing to find. I don't know how to pull the codes out, hopefully someone will respond on finding the trouble codes, since that will make it easier to narrow down the fault. Or use the Search to look up how to pull codes from ABS.

The system does not need all wheel speed sensors to work in order for it to run through its self activation test. It can see the car rolling by monitoring any of the other working sensors, and then it will run the pump briefly.