Once upon a time, plugging in the battery backwards wouldn't kill anything.
The +ve is usually the centre lead, so plugging it in backwards just means that battery negative is applied to the receiver decoder signal output, not normally damaging. Plugging the battery direct into the servo would just apply battery -ve direct to the servo signal input, again not normally damaging.
To cause wires to catch fire and plugs to melt requires amps and amps, so either there's a reverse protection device crowbarring the battery or it's the servo mosfets that have reverse conducted somehow. In the second case, fiddling with the pot isn't going to do anything. One should be able to get the servo to move a little by waggling the sticks even with a squirrelly pot. So my guess is that the FETs are friedor there is a fault in the servo signal path.
Of course, if you have a connection system where the +ve and -ve are on the outside wires of a 3-wire system, and a connector that can be reversed then that's a fire just waiting to happen!