Could someone break down why the magnetic compass reads accurate on all turns-to-headings, regardless of geographic position, when performing rudder-only turns with no bank?
A magnetic compass can point out direction, provided it is free to move in all horizontal directions and not exposed to any G-force other than Earth's gravity itself.
Sailing vessels operate gyroscopic hinged compasses to facilitate that. Ships can use those, because G-forces are seldom a factor at the slow turning speeds involved.
As airplanes are unlikely to operate at a continuous tilting angle, like on a sailing vessel, but are more subject to G-forces, they use simpler magnetic compasses mainly as a reference. Those are usually semi-spherical compasses, that allow for some tilt, but not for G-forces.
Direction on airplanes is therefore primarily read from a heading indicator, which is not a whole lot more than an axial readout surrounding a free hinged very fast spinning object, the axis of which points in a preset universal direction.
As long as it spins, this axis will point in that direction, indifferent to not only tilt and G-forces, but also planet rotation and even the Earth's orbit around the Sun. As an airplane's flight path occurs along the surface of an in itself spinning planet, this requires the heading indicator's spinning axis to be regularly realigned, using a stably functioning magnetic compass as reference.
The notion that magnetic compasses would not show error in skidding turns is not correct. Operating a magnetic compass outside it's tilt tolerance will make it show erroneous read out, as will exposure to any G-force other than earth's gravity.
The foundation of the myth may be that many compasses are seated on a very fine pin and rotate along the plane of the horizon.
Banking or manuvering may cause one of the ends to stick. Also, banking moves the compass out of plane so "North" may become "up" or "down", again causing error.
But yawing can also cause the rotation to "overshoot" its proper reading and spin past it. This is one reason to fly a rectangular or box pattern, allowing instruments (and our senses) to settle unaccelerated for a bit.
The heading indicator instrument is there to give more reliable readings while turning. It uses a rapidly spinning gyro as its reference to directional change.
Interestingly enough, the heading indicator is subject to precession errors, and must be regularly reset with the compass in unaccelerated level flight.