[Chas]

I placed the cabinet in the bathtub, sprayed it down with Spray Nine and dark brown liquid looking like dark chocolate oozed out and continued after numerous rinses and reapplying.

After a number of hours of polishing using Automotive Clear Coat Finishing polish it recovered from the cleaning process looking very close to what it originally looked like when new in 1936 (I hope)

Factoid: Bakelite is not an indestructible plastic. It can be dissolved using caustic chemistry, particularly strong household cleaners.
Although this is not 100% true for some Bakelite formulations, exposure to sun and household contaminates alters the surfaces. Therefore, the same radio may hold up to a strong cleaner but from another environment will be dissolved. What was actually happening is the top, gloss layer, of Bakelite was removed. The smell of phenol is most obvious and is often confused with nicotine tar..
Some Bakelite have a course filler to keep it stable from shrinking. The outer layer is caused by the filler migration away from the mold. It is not very thick, wash it off and the filler is exposed. This is very difficult if not impossible to polish.
Consider using a far gentler cleaner such as Murphy's Oil soap or the Go-Jo method. Either way the surface will be clean without dissolving the Bakelite.
YMMV
Chas