So this is probably a dumb question...but is the clock at all accurate? It says it stops periodically when there isn't sufficient fluctuation in temperature so do they manually correct it after?
I have an Atmos clock that's probably 40ish years old, handed down from inlaws, and while I'm sure it needs some service to get it up to snuff, it looses about a minute a month, give or take.
I don't see how the winding mechanism should impact its accuracy. Obviously, it it runs out of energy, it needs to be reset once working again. I'd think that goes without saying.
However, the mechanism that makes the clock tick-tock once per second is, again, separate from the winding mechanism, and I don't see why you'd have to (as another poster commented), "choose 1".
The article mentions that it has stopped a few times, which would affect accuracy.
The Clock of the Long Now operates on similar principles (temperature difference), but separates timekeeping from the display mechanisms in part to maintain accuracy.
And also so that the power required for the display may be redirected to the more critical timekeeping function if necessary. If you only have enough power left to keep time for the next month or to power the display for the next day, you turn the display off and hope that more power is added to the system during that month.
I own a similar clock (the Atmos clock mentioned in the article). It loses several minutes per month. FWIW, I live in Los Angeles and it has been the same regardless of time of year over the past decade or so. It's great that I don't need to wind it, and it's mesmerizing to watch, but it's not the most accurate clock in the world.
If it is consistently slow (or fast) then it sounds like it should be correctable; most good clockwork mechanisms have some way of adjusting the beat rate.
Indeed Atmos clocks do have a speed adjustment, though they are very sensitive. I'd contact a professional since any repairs to them get expensive very fast.