I know that mechanical watches have a spring that they wind to store energy, and un-winding the spring produces energy for the watch. But a spring produces a lot of force when it’s very tightly wound, and very little when it’s almost completely un-wound. So how does the watch even that out with high precision?
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A mechanical watch doesn’t keep perfect time. A well regulated wrist watch will be +-2 seconds a day. Even up to 10 seconds off isn’t bad, depending on a particular movements specifics. I think there are some marine chronometers are accurate to .1 seconds a day.
To answer your specific question, all mechanical watch movements have an escapement, part of which is a spring. This is the part that “ticks”. There are various tricks watchmakers use to even out the ticking. There are even movements designed with “constant force escapements” to address this concern directly. But even if the movement isn’t dedicated to this directly, modern watch springs have a pretty flat force curve for the majority of the time the spring has enough energy to deliver power to the escapement.
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