This isn't the definitive answer by a long shot, but my understanding is that servos are either limited in how far they rotate if you want to control position with any accuracy, or modified/built to rotate freely in which case you only control their speed and direction of rotation, not the position they stop at.
A stepper motor will allow you fine control of how far it rotates in one direction or another, with no limit to the number of turns, but it will not center to a desired position or provide high torque in maintaining that position against other forces. You just tell it X clicks clockwise and it tries to do that, with no guarantees.
In the end, you are (I think) trying to create a closed control loop so that some motor provides torque to rotate something until some sensor indicates a desired position has been reached. A stepper motor by itself, or a continuous rotation servo, do not provide the sensor part of the loop, while the sensor in a standard servo does not support rotations past a certain range.
So you need an alternative sensor that tells you where in the rotation you are, and to generate a control signal to the motor (of whatever type) accordingly. Servos use potentiometers to sense position, so perhaps a pot or two with weird tapers might work, or the good old optical encoders from a mouse with modified discs?
Having said all that, if anyone knows of an off-the-shelf servo that behaves as you describe (since it's not inconceivable to make one, I just don't know of any), I'd love to save the link to the vendor for future reference!