But then the question is, does a servo motor allow for manual control when it is not operating?
For typical R/C servos, the answer would be "no", as long as power and position signal is applied; if the power is removed (and the positioning signal - you should never apply a positioning signal to a servo without it being powered; it can damage the servo - some servos have protection against this, but not all). If there is no power supplied to the servo, the servo can be "back-driven", but this is bad for the internals of most servos (stripped gears being the most common).
Won't it just lock in a particular position, preventing the user from manually using the switch?
Only if power and signal is applied - see above.
Is there a particular type of servo that I should look for? Or maybe this approach is wrong to begin with?
I would think if such a device existed, it likely wouldn't be a standard R/C servo (or use the same signals), and it would likely be very expensive.
Back in the day, when televisions had knobs and remote controls came on the scene, there were motor controlled channel changers; they allowed you to change the channels on the TV, but you could still manually switch the channels if needed. I am not sure how such devices worked, but I imagine that they had some kind of integrated electro-mechanical (or similar) clutch that would engage the switch when turned on, then disengage (to allow manual control) when they turned off. In effect, probably something similar to how the starter motor on a car works (the gear on the starter motor only engages the flywheel while starting the car; there is a solenoid that controls the position of the pinion on the motor - though some motors use a completely mechanical method as well).
So - perhaps look into how these systems do it; I would have to say, though, that this would be something you would have to build yourself, and it wouldn't be easy nor inexpensive. There may still exist motorized multi-pole switches; but again, they are not likely to be inexpensive (if they can be had at all).
I am curious why you are going for such an electro-mechanical method in the first place, and not simply using a digital solution of some sort...?