I would say that the acid probably leaked into the case by soaking
around the metal battery contacts and has likely corroded the wire
where it connects on the other side. The wire is probably corroded
almost all the way through. Enough connection remains to power the clock
as it requires very little current to run. Any more current drawn
reduces voltage.

Just a guess of course but I have seen this many times with other
equipment in my shop.

Others may have other suggestions of course.

Good luck!

Andy