I don't know how the stock chain counter works, but I was able to fix the one on our boat (I don't think it's original) by duct taping a magnet to the windlass.  I'm unsure if the original magnet fell off or just lost its magnetism, there's just a tiny rusty insert where it should be. The sensor was fine.  What makes you think there is a short in your sensor?  If you're just looking at the output of the sensor with a voltmeter while the windlass is running, it's possible the pulse is happening too fast for the meter to react.  You might want to try using an oscilloscope if you haven't already.

I think it's unlikely your new sensor will hurt anything, since it will be providing the same signals as the existing one -- just with opposite duty cycle.  You may find that it works perfectly if the display is edge-triggered rather than level-triggered, as I suspect it is.  if you do find you need to invert the signal, it should be possible by adding a relay. The exact way to wire it up will depend on specifics of your system.

Good luck,