I found this similar question with an answer explaining that wheeled transport was banned in the early 17th century for complex political reasons. So previous to that, the roads were capable of supporting it. Now I kind of wish there was a plotline with Kagome (the show's protagonist, for non-fans) getting in trouble with the authorities for having a bike!

The roads built during that time probably wouldn't have been capable, though.

Expanding in many areas on ancient foot trails, the arteries were first constructed to secure Tokugawa’s power, allowing easy transit for officials and a way to monitor the populace. Although beautifully engineered and referred to as “highways,” the tree-lined paths, which were mostly of stone, were all designed for foot traffic, since wheeled transport was banned and only top-ranking samurai, the elite warrior class, were legally permitted to travel on horseback.

Edit: I have been told to ping u/asiaexpert for writing the answer I referred to