What batman is suggesting looks pretty nice - never have seen that before.
The difference in the video from the motorcycle cables and the cables in our mid-50s Fords is that the cable has a solid core, or wire, that slides in the sheath. The cable shown in the video is made to be pulled in one direction while the other end retracts from something spring loaded or under some other pressure. (like a motorcycle caliper, etc) Basically, the wire strands inside the sheath in the video are being pulled in both directions...
For our Fords, the issue lies in the solid core being pulled and PUSHED from the same end. This makes the lubrication issue all the more important. For each and every one of these cables down through the years I have always had to pull the cable out of the car to clean and lubricate the solid core. I recommend remove the old black insulation, cleaning any and all rust/corrosion with a wire wheel, and then lubricating both ends with a light oil such as mineral oil. In addition, it is helpful to liberally coat the sheath with light oil and work the oil into the solid metal core by bending/rolling the cable back and forth. (by rolling I mean to make a circle with the cable like you would a drop cord) Let the cable then hang vertically and drop some light oil on the solid wire core at the top and allow the cable to sit overnight.
Work the cable back and forth as it frees up. Once it is free, you can wipe all excess oil off the sheath and use long pieces of shrink tubing to re-insulate the cable.
this method has worked for me for the past 30 years or so...