I’d start by running a pair of wires from the car side of the plug to the sensor on the opposite side, if you get readings you know the wiring is ok, if you don’t its either the plug or wiring (see below)
You say you’ve measured the voltage from the sensor, it’s a weird one, you may see something on and oscilloscope with the disc turning, but a normal multimeter will be all over the shop
It’s an ac (inductive) sensor, so it generates its own voltage so in effect you won’t see anything constructive on a multimeter, you may well see a trace voltage from the abs pump side which can be an indicator the wiring is ok...
the only true way to check the wiring is to unplug at the abs unit end plus the sensor end and add 12v and earth and add a h7 bulb at the other end of the circuit, if the bulb lights wiring is ok... a resistance test is fine for a quick check if you short out the plug end... but... one strand of wire will give a positive result on a resistance check, whereas a single strand of wire on a load check with a h7 (55w so just under 4A) will generally fail.
If it fails take a single core extension and run it down the side of the car and test each one individually, if you have wiring diagrams with plug pics on find the middle connections and test either end the same way
This one was a freak one off!!! I have had one instance where it was the plug on the end and me putting link wires in for a load test hit a different part of the plug and gave a good result, but the pins were damaged where they made contact with the abs sensor plug pins, sucked that one as it threw up a massive false positive and needed a new plug but all appeared good, sent me off on a right merry goose chase!
If the wiring is ok with a load test and swapping left to right and vice versa doesn’t work it then either plugs (incredibly rare!) or the abs module itself (more likely)
If it’s an abs module it’s sometimes possible to buy just the module and unbolt/bolt back on... if your unlucky it’s a pump and you’ll need to bleed it all
I suspect yours will be either a gap issue or a sensor (I’ve had two brand new sensors that looked identical on a vag, and one worked one didn’t on a Passat, both sensors were new!) or maybe a pump