I bought two servos online used, and one of them works great. I can swap it between the steering and throttle slots in the ESC, and it works like a champ. The other one I noticed has a SMALL cut in the black wire, which initially was so slight I thought it was just in the wire casing. On closer inspection, it looks to may have bitten into the wire a bit, nut not all the way through. Could this cause the servo to just not work at all?

That ground wire is probably cut through and not making contact so it won't work.