As mentioned, those clocks had a pair of contact points (similar to distributor points) that energize the movement. With age, they get corroded and "burnt" just as the distributor pionts used to, so cleaning and dressing them can usually put things back to normal. Also, cleaning and lubricating while the case is apart is a good idea too.
There was also a "rewinding" procedure that I read about somewhere. It could well have been in a Chrysler manual from back then, but I don't recall exactly. What it amounts to is somewhat time consuming in that you have to have the battery cable disconnected when the clock is installed. Then, you touch the cable to the battery post, which should cause a spark, and then let the clock run from that electricity. Then you repeat that procedure several times until there is no spark when you touch the cable to the battery post. That was supposed to be the "correct" way to re-hook-up the clock if it ever ran down from a dead battery. Seems like each "click" was about 15 minutes?
Most mechanics just put the cable on and let things take care of themselves. While they could keep pretty good time, it seems that no one really cared just how accurate they were back then, also knowing that at some point in time, they'd stop working.
The clocks are also self-regulating too. When you adjust the time forward or backward, the clock will alter it's regulation slightly. Remember, those clocks run on DC voltage instead of the self-regulating AC voltage as a house clock does.
Putting a quartz movement in the clock case is an option too. A much more accurate movement that probably takes less power to run too. If I went that route, I'd make sure that it looked just like the original (hands and their finish) so no body might know unless they were told about the conversion.
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