The wire to the pads is the brake wear indicator - nothing at all to do with the ABS. The wire is attached to a probe inside the pad. When the pad gets worn down to minimum, the probe earths through the brake disc and you get an alarm, which comes via the "handbrake" light, If the wire is cut, then obviously there will be no earth when the pads wear out, and no alarm.. but the car doesn't know whether the wire is cut or not. Not having it makes no difference to the brakes or the braking.. apart from you'll have no future warning that your pads are low.
The ABS sensor measures the wheel rotation at each corner. If one wheel is spinning at a different speed to the other one on the same axle. the car's brains think that wheel is locked, so the ABS comes on, to relieve brake force at that caliper.
The ABS sensors measure wheel rotation all the time, so the sensor is always feeding back a signal, If there is no signal when you start the car (the sensor is out of contact) then you get an ABS warning and ABS is disabled (brakes will work as non-ABS assisted brakes). If there is a sensor signal then that's cool.. but as soon as the wheels start to rotate, the ABS computer will be comparing the signals from each wheel. If one sensor is lazy or sending back poor information, or no information, then the ABS warning will come on and ABS will be disabled.
You have a sensor that is not communicating with the car. Replacing it is not too expensive and relatively easy, if the sensor bolt hasn't seized up or rounded off (it's an allen head, so they fill with water and just rust out the centre where the key goes).
Once you have the allen bolt out, you need to prise the sensor blade out of the caliper. Pull it out vertically.. it's like a little spade so if you bend it in the wrong direction, it will snap. The broken piece can be tapped out from inside the hub.. but use a plastic prong, pick or fork.. anything magnetic (steel screwdiver etc.) can de-mangnetise the tone ring.
The tone ring aka wheel speed sensor ring is a magnetized nitrile rubber ring, brown in colour.. on the back of the wheel bearing. It sends a magnetic spike to the sensor every time it passes from "north" to "south", so that the sensor knows how fast the wheel is going round.. by measuring how often it gets this signal. You don't want to ginger that ring up.
Ralf S.