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Before the invention of the blast furnace, metal was so expensive, a bike would have cost more than a horse. I read this essay by a medievalist mentioning that the metal in a bit and bridle was worth more than the horse, and the horse was worth more than a house.

Incidentally, this is hooey; metal was of course _more_ scarce, but people of all classes had knives; you can see metal-bladed farming tools in museums with a lot more metal in than a horse's tack, farmers not being noted for being the most wealthy part of society; and so forth.

1) Surviorship bias. A tool made from wood, bone, or horn is less likely to survive long enough to make it into a museum collection. 2) Preservation. Metal tools were expensive, so extra care was taken with them. This made them last longer. They often got passed down through generations.

Making a pushbike out of wood is theoretically possible. It would be heavy and uncomfortable, and making a reliable powertrain out of wood is a tall ask. Plus, vulcanized rubber wasn't available for tires until the 19th century.

It's practically possible; they're made of of bamboo today, and have been since the late Victorian era. The early dandy-horse designs were largely made of wood, too.