I have recently repaired my dash clock, same as your Mach1 clock. I bought the car last year and the clock was not working. I applied a small amount of WD40 to the mechanism and manually turned it quite a bit. Reconnected to power and it ran, but very fast, extremely fast, around 6 hours per day, which is useless really, and this is due to the hairspring being gummed up and wet with oil, NOT just the adjustment lever.
These old clocks are mechanical, and the winding mechanism is a contact array that is triggered every few minutes. The mechanicals are fairly robust. The electrics are pretty good, considering they are old, and in comparison to Chevrolet clocks of the same era, and later models too.
If the coil burns out it is unlikely you will find a repairer interested in installing a new coil and that is the time to replace the movement for a quartz unit.
I agree with the advice given to apply watch oil or light oil to the gears and the main base plate, little by little. AVOID spraying oil on the hair spring. the hairspring is the very thin coil spring under the balance wheel (balance wheel is the only thing that spins back and forth)
You can manually push down the small contact and get the clock running. If it runs on until the contacts close it is good to go.
Re-apply the power connection and watch it for a few cycles. If it resets and ticks over properly, then start timing it. If it runs grossly fast then you need to clean or dry the hairspring. I took mine to a local backyard watch repairer who cleaned the hairspring with some high octane fluid. I would think either brake clean, or methanol would work too, maybe rubbing alcohol. You cannot use Q-tips or anything like that, it will foul and stick in the tiny hairspring . Squirt it using a syringe or something.
That is the difficult thing, you need to oil the working parts without getting oil on the hairspring, and then clean the hairspring with 'alcohol' without drying out the oiled mechanism. Do you follow this?
After that, timing and adjustment is simple. if the clock is running reasonably close to correct, say 1 hour per day fast or slow, just keep adjusting it every day. As Bartl says, the time-setting gears are also connected to the 'fine adjustment lever' of the movement, and will eventually set the clock accurately.
I have replaced a 78 Corvette clock movement with the same quartz movement and they work fine. I was not impressed with the retaining discs/clasps supplied in this kit and urge care when or if reassembling the clock. As I implied, the Mustang clock sare pretty good, and their electric contacts are usually ok with very minor touch ups. Chevy must have gone cheap or got bad batches on their clocks. Almost every single corvette clock from the 60's through the 80's do not work and don't respond to a contact clean - they are burned through or fused.
So give it a go on the refurb, it cost nothing but time and you always have the quartz clock as a back-up plan