that's the centrifugal switch
The switch engages and disengages the run or start windings as it revs up. That motor has no brushes. Only a motor marked AC/DC has brushes, called a universal type motor. Hand held power tools, drills and circular saws have them. Direct drive saws,.... el cheapos have them also. That motor is as bulletproof as they come. Just belt it up and run it!
That restoration look really nice! I owned a Craftsman 100 saw from the 60's with that same look only mine was a gold color. It ran for 45 years, never quit, I just wanted to part it out for the table.
The answer to your question will only be as detailed and specific as the question is detailed and specific. Good questions also include a sketch or a photo that illustrates your issue. (:< D)