I can't tell you how many Ford clocks from the mid fifties up through my 67 Mustang that I have fixed over the last 50 plus years using the same basic procedure as Jim has noted. My "burnishing tool" of choice has always been a piece of 320 wet/dry sandpaper, folded over so that when passed between the points, it "burnishes" both points. Over the years, I have seen this to be the most common problem causing clocks not to work. I figured this out in the early 60's for my '57 Retractable (wanting everything to work correctly on the car), and when I was showing it during the 80's, other club members couldn't believe that my clock functioned. I resurrected, and saved from the scrap pile, many 57-59 friend's clocks during that period. I finally had to do it again to my 57's clock in the 90's, and probably will again once I get it back on the road again, since it has been sitting for about 8 years now. I have also applied this same fix to my 59 Retractable and Ranchero, my 57 and 62 T-Birds, and my 66 Rallye Pac and 67 clock with successful results. The "fix" doesn't take near as long as the R & R.
Bob