Good investigative work Denys, Ray, Rob. I watched all vids and learned a bit. Now we know what we are dealing with. It would be a good exercise to find out why there is only 7V out not 117V.
The first time I came across this kind of technology that uses the principle of saturating the transformer core to alter its output was in an Ericson PABX telephone exchange, they used such to alter the audio level as a form of automatic volume control. I came across this back in 1975, then there was little information to be found on the applied principles, I came across a telephony book produced by the then PMG that instructed on several techniques using transformers, aside the saturated reactance, there was one that allowed three parties to converse on two wires without any crosstalk, this was achieved by using center taped transformers at ether end, the center tap would go to ground via another transformer, this was called the Kylo system, (not sure on the spelling) using the balance and unbalanced techniques.
I had not come across ferroresonance but it still uses the core saturation principle that alters the the field flux by pre-saturation of the core, the difference is that in the ferroresonance system it allows the field saturation to reverse allowing a greater range of control, (Very Clever)
But it should be noted that this is at the cost of sine wave distortion creating second and third order harmonics, this can be detrimental to any other transformers connected downstream creating heat but a small trade off to obtain regulation.
Lots of clever things had been developed in the olden days to mitigate issues in design, these things are now mostly lost in the face of modern techniques and solid state componentry.