Actually, the transformer visible in that picture of the internals looks like a ferro-resonant constant voltage transformer. Note that there appears to be core between the two winding sections. This was pretty typical for CV stuff - Sola is one brand name I remember. I'm not sure what the gizmos on the plate on the left are but looking again, they are probably terminals of a capacitor clamped down and sticking out the underside of that plate/chassis.
What is under that plate presumably holds the key. There are indeed usually capacitors in ferro-resonant supplies and caps do not age gracefully.
(Of course, it could be good now that you've let the smoke out. )