Got a strange problem and hoping your engineering background could help.
I have a Seiko chiming clock (with pendulum) that runs on a single C battery. It's developed a problem when I replace the batteries with new alkalines or regular carbon zince batteries, it runs slow. I'm talking losing a minute an hour! However, whenever I use an old battery or let it run down, the clock keeps time!
What do you think the problem could be? Is the higher voltage of new batteries causing some problem with the motor (over torque)? Like when you dump the clutch on a car, you get wheel spin and go nowhere.
I can't think of a better reason. Let me know your theories!
I was reaching for ideas when throwing out battery connections as the culprit. I truly have no idea how a quartz clock can lose so much time and go back to accurate with a nearly dead battery. You seem to have ruled out battery chemistry by trying multiple types.
My only experience in this topic is from opening a single cheap battery movement. The quartz crystal was visible along with an epoxy coated semiconductor. The only other major component was the "motor" that appeared to operate more like a stepper motor than a DC motor. It likely has a very small number of distinct poles and operated as a low-resolution stepper motor. The armature would rotate a specific angle once or twice per second under control of the processor. The armature should rotate as long as the voltage is above a minimum threshold.
There seems to only be a few cases where the clock could lose time. The voltage is so low that the motor stops operating. Or something is causing the oscillator to not be able to keep accurate time. Or the grease on the internal gears is thickening.
I have a single data point referring to the 10-20 year useful life of a battery clock movement. I received a wall clock as a gift in 2002. The movement failed and was replaced about 3 years ago, so around 17 years of life. It is possible that modern movements are susceptible to the "race to the bottom" where cost is the only criteria that matters. I am hoping to get a few more years of life from it.
I am happy to report that all of my clocks have an accuracy of better than a minute per hour. LOL Most run with an error of about a minute per week. The "silent" desk clock is accurate to about a minute per year and never needs winding.
The internal operation on most of those style clocks is a mystery. I believe they operate similar to stepper motors that move once or twice per second. That is why many of them make ticking sounds. The oscillator is only acting as a counter to decide when to apply the next tick.
A crystal oscillator should be fairly tolerant of voltage changes, while a simple DC motor would slow down as the voltage drops. The pendulum drive is independent of the oscillator, and it should reduce amplitude as the voltage drops.
I can only make a few random guesses for why the clock slows down with fresh batteries. My first guess is that the old battery was corroded, and the acid actually makes a better connection than a clean new battery. Or else your "new" batteries are not as new as expected. They can go flat sitting in a drawer for a few years. You could try measuring the voltage at the terminals inside the clock.
My next opinion is that quartz clock movements are somewhat consumable and easily replaceable. Chiming movements might cost around US$30-40. A new unit should give the clock an additional 10-20 years of life.