If your light stays on from initial start-up, it has failed the self test. At this point the only way sensors can be involved is if they are open circuited (disconnected). Dirty, fouled, missing teeth, none of this matters to the initial self test, only continuity.
If the light comes on after moving then something is almost always wrong with the sensor signal quality.
The advise about the OVP relay is good. It is the number one reason for continuos light operation. next is the pump relay or disconnected sensor, and lastly is a bad control unit itself. Oh, and yes lack of alternator output will keep ABS from turning the light off
Bosch Master, ASE Master, L1
33 years MB technician