Yes, it's called a saturable reactor [0] and can be considered as a very simple type of magnetic amplifier. You control the impedance of the reactor using the extra winding, conceptually similar to how you control the conductivity of the pass transistor in a solid-state voltage regulator. You can find them in old Tektronix oscilloscopes, here's a long (70 min.) and comprehensive video [1] on the design of the Tektronix 555 power supply, including the operation of the saturable reactor voltage regulator.
Yes. Saturable core reactors have a long history of industrial regulation of heavy duty motors, furnaces, and the like. I first learned about them from old cranes in seaports.
For some applications size and weight aren't a big deal, but durability and overload tolerance are, so I can imagine they are still used out there somewhere.
