I haven't used the BG6500 for a while, which may be why the drive belt for the servo skipped a beat when I tried it again, and forced me to open it up to get that reattached.

Now I find the left channel is not performing as it should, much weaker signal. I've had it up again, and have checked and cleaned the muting switch without things improving. Everything's attached as it should be, so now I'd like to remove the muting switch, and just want to be certain as to procedure.

Do I just unsolder its four pins into the board and then remove it?

Try to move up the PCB with the switch, up until its free of the lever that actuates its, then check with an Ohm meter  that you have no connection between the legs, if thats OK its not the switch. If the switch is bad it will not mute.