going by the limited view in your photos
its going to take more details to reverse engineer your control box
the control in the video is easy in conparison as it has the spare relay on the combined BOB and 3 axis driver board
and labelled terminals on the PWM spindle speed control
Attachment 431216
in your control box
you have a very simple BOB
and 3 individual stepper drivers
Attachment 431218
your spindle speed contol looks like a modified power supply with a remote potentiometer
Attachment 431220
if its as I expect
you may be able to connect an unused signal from the BOB to a relay with a built in current amplifier
the relay then switching the mains supply to the spindle control
John