Swinging chokes do NOT work as reactors. I don't know the physics behind that one though
The swinging choke is a saturable reactor. It is designed to saturate with DC flowing through it when used in a choke input supply. At low currents the inductance is very high, to maintain critical inductance so that the filter doesn't become more like a capacitor input filter, which would cause the plate voltage to soar at light loads. At heavy loads, the inductance drops drastically, but less inductance is needed when the supply is delivering higher current. Sometimes a second, smoothing choke is used, since the inductance of the swinging choke may not be enough at high current loads to adequately filter out the ripple.
The advantage of the swinging choke over one with a constant high inductance, is that the swinging choke tends to be more economical, smaller in size, and to have lower DC resistance than would a higher fixed inductance choke with the same current rating.
Because the swinging choke saturates at full plate current to drastically lower the inductance, it would make a very poor modulation reactor.
The swinging choke is constructed very similarly to the smoothing choke, except that the gap in the core (there to reduce saturation from the DC) is much narrower, to maintain a high inductance at low current, but the increased saturation causes it to drop to a low value as higher current flows through it.
Modulation reactors tend to have relatively high DC resistance compared to smoothing chokes, but the DC load is constant when running AM, so this doesn't cause the plate voltage to vary. It may actually give a slight edge to the modulation capability when using a common power supply for modulator and final, since the voltage drop through the reactor results in a slightly lower plate voltage on the final as compared to the modulator.
In the old class-A Heising circuit, a power resistor was usually inserted in series, between the modulation reactor and rf final, to drop the DC voltage to the final, allowing 100% modulation capability. The resistor was by-passed with a capacitor so that the resistor would drop the DC voltage but not the audio voltage.