So...
Maybe all the clock needed was oiling. I fully removed the mechanism today and inspected it (ingenious, in the traditionally understated 1950s analog way).
After studying it for an hour or so, and being more impressed the more I looked, and after I gently scraped the contacts (which looked to be in pretty good shape), the WD-40 had penetrated its way everywhere it needed to, and it was ticking somewhat reliably. Back into the car it went, and when I reconnected the fuse, I heard the happy little whirr of the winding mechanism, and now the car ticks like a time bomb. I'm going to leave it overnight, and if it's still running tomorrow, I'll put the securing bends back into the case and make all fast.
A note of history. This clock was made, according to the stamp on the back of the case, by the New Haven Clock and Watch Company, which was formed in 1853, and remained in business until filing for bankruptcy in 1956. The company was finally broken up in 1960.
See yesterday's post for questions.