Hi
No, the tone and volume controls do not have motors on them. Each does have a gear on the end of each control shaft; this gear contacts teeth in the tone and volume thumbwheels, respectively. This gearing is what makes the controls operate.
Sometimes teeth break off either the tone or volume thumbwheel, or both. Where the teeth are broken, the thumbwheel cannot turn the control up or down. This can only be remedied by finding another thumbwheel from a junker set. The thumbwheels are made of Bakelite, and the teeth on these can be broken easily if you try to force the control.
Now, as to your other question.
Is it worth fixing? That depends. If you plan to fix it up to resell, then no. If you want to keep it and enjoy it, then yes. One man's opinion...
Finally, please do not plug it in and turn it on again until you have, at a bare minimum, replaced the set's electrolytic capacitors and tested the tubes. More info on that subject
here.