Ok the fact that it burns all the time and hasn't killed the battery is Important.
Ok, got it, The design of that dash makes the light burn if there is no connection to the regulator.
So yes with that wire off it will charge and work normally.
Yes this is rescinding my other post that your altanator or regulator is bad because I didn't realize that the only effect was a light was on.
Just for a note to all:
Quick way to find out if your alternator is working, with the tractor/car/truck running take a screwdriver and touch the front or back of the alternator shaft, if it sticks magnetically it's working.