There are lots of clocks made with wood gears, including almost all clocks made in what is now the US from 1600 to 1850 or thereabouts. Wood has some significant advantages in precision — although it unavoidably swells and shrinks substantially across the grain, as humidity and temperature rise and fall, its variation in length
along the grain is much smaller, in the 1ppm/° neighborhood of Invar, an order of magnitude smaller than brass or steel. This makes clocks timed with a wooden pendulum an order of magnitude more precise than clocks timed with a plain metal pendulum; this knotty problem evaded generations of clockmakers until finally falling to legendary horology hacker John Harrison and his gridiron pendulum.
Imprecise gears don't keep bad time; what they do is increase friction and wear out. Your gear ratio isn't going to change from 51:20 to 51:21 because the gears are soft, flexible, and vary in size with humidity, unless they vary by so much that the gears come out of mesh. As long as they're in mesh, each tooth on one gear is followed by precisely one tooth on the other. Rather, what will happen is that the depthing of the gears needs to include extra backlash to allow for the imprecision, and that imprecision causes deviations in the pressure angle and the like. This is a terrible problem if you are gearing down a steam engine to pump water out of a coal mine, but at the much lower loads encountered in horology, it just means you need to wind the clock more often. Chris on Clickspring has convincingly shown that even the primitive triangular tooth form used in the Antikythera Mechanism is quite adequate.
The cycloidal tooth form used in modern clocks was only invented in the 1600s to 1700s, while mechanical geared clocks built in the 1300s still exist today, like the one in the Sarum Cathedral, which is iron: https://www.youtube.com/watch?v=KbJb92H5gy8 — the clock uses a square tooth form with rounded corners on the large gears, meshing with smaller pinions whose "teeth" are round dowels. (This part of the mechanism is believed to be original, but the escapement has been replaced.) By contrast, the tooth form depicted in the 1364 Il Tractatus Astrarii is triangular, like the Antikythera Mechanism. The clock at Chioggia, also from 1386, seems to be made in precisely the same way from what I can make out from these photos: https://www.adjora.it/altro/curiosi/vari/orologio-antico-Chi...
Note that the Sarum clock is held together with tusk tenons like those in the 7000-year-old well discovered in 2012, but in wrought iron rather than wood. That's because in 1386 the screw was about 2000 years old but not yet in use as a fastener.