This style of radio was made in the mid to late 1920's. Earlier these tuned-radio-frequency sets had three dials, one for each tuning condensor. Then towards the end of the TRF era, manufacturers tried to simplify tuning by ganging them together on one dial. The sets didn't track perfectly thus the "antenna" control on this set which allowed tweaking the alignment of the first RF stage. People used sets like this to hear stations all over the country.
I found more pictures of the set here:
http://radioattic.com/item_more.htm?radio=0038243
It would appear that the small inner knob on the "volume" control turns the black cam on the back of the rheostat. The missing rheostat controls volume by the voltage allowed to the tube filaments. The black cam would have actuated a switch to turn the "A" power on/off. The cam appears to have a metal section on one of the indents. That section is probably connected electrically to the rest of the rheostat. A leaf spring such as one from a relay or a headphone jack or from the guts of an old telephone could be fashioned to ride on the cam to make an on/off switch. Can't tell but bet that it's a turn-turn switch that goes on/off/on/off as you keep turning in one direction.
The rheostat resistance winding missing complicates things. Finding a duplicate with concentric power switch is going to be impossible, if not worse
. My first thought would be to make some brackets to mount a new rheostat behind the old and put a lever arm on the new one like the old one has, and connect the lever arms together. Not impossible to do, would take a little time planning and bending up some brass strips for brackets (try a good model train shop for brass or aluminum.)
Not sure about the resistance value of the rheostat but will do some checking. Someone else may have an idea.
They used storage batteries for the A power on these sets because the tube filaments draw so much. For testing you could use a 6-volt lantern battery but it wouldn't last too long. The B supply would last longer and can be a series of 9-volt batteries snapped together + to - . A wire can be captured in between snaps at the appropriate points for intermediate B voltages like 22.5 and 45. The C supply can be made up from AA cells and will last forever. For long-term use you'd want to look into purchasing or building a power supply for the A and B voltages.
You can use any modern PM speaker with any audio output transformer between it and the radio for testing. One out of an AA5 radio would be fine.
Very nice looking radio in great condition!