Saying that the bicycle "doesn't seem like a very difficult work of imagination or engineering" is a good example of how easy it is to take technology for granted. There are a couple of different "ingredients" which made building "good enough" bicycles possible as well as which made bicycles worthwhile enough to become popular.

You could easily imagine building a "bicycle shaped object" almost any time since the neolithic, but the question is how useful and practical such a device would be. Ironically, mountain bikes using active suspension and large tires would be particularly useful in a pre-industrial setting, able to cross long stretches of unimproved paths even through challenging terrain, but they are fiendishly modern and require pretty recent technology to be manufactured and even maintained.

The earliest "proto-bicycles", otherwise known as "velocipedes", had no cranks and were primarily for coasting. They were made mostly out of wood, and one could imagine building something similar at much earlier times throughout history. While this is true, the velocipede was mostly a curiosity, not really much of a practical device, and it's only interesting in that it served as a genesis point for what would become the bicycle. One of the biggest problems of the velocipede was that it gave a very rough ride, despite being limited to fairly slow speeds, due to the fact that it used wooden wheels, which gave rise to one of the common nicknames of the device: "the boneshaker".

The problems with a pre-bicycle bicycle is that it would be a terrible riding experience, and would have limited usefulness. Using a wood frame would limit its longevity. Bicycle frames need to be incredibly strong, unusually so compared to lots of other devices. There's a reason that wood bicycle frames remain a niche curiosity even today, wood just isn't strong enough due to its grain. But that's not the only problem with proto-bicycles. Another major problem is rolling resistance. If you have just a simple axle and wheel design, whether it's metal or wood the amount of friction involved will thwart the efforts of the rider and make for an unpleasant experience.

I would consider the absolute minimum feature set of a bicycle that would have more than niche usefulness to be a welded tubular steel frame and ball bearings for the wheels. Both of those are 19th century (or nearly so) inventions, which makes the bicycle a solidly 19th century device. More reasonably, things like rubber or pneumatic tires, wire spoked wheels, and geared chain drives (all the components of the "safety bicycle") along with increasing urban populations are the major ingredients for bicycle popularity, and those too are 19th century changes.

So, realistically you don't have the core enabling technologies and societal/economic changes necessary for bicycles to be successful until the 19th century and you can't easily push that forward very much, certainly not by centuries. You could imagine creating some type of human-powered vehicle with pre-19th century technology. Maybe a tricycle with leaf spring suspension? However, the designs become more complicated and difficult the more technologies you exclude. And realistically I don't think you can get a reasonably useful vehicle without bearings.