I think a bike could have been possibly made with a limited range without needing a chain drive by incorporating a fixed length spool of strong, thin string into the hub of the front wheel up to another spool attached to the handle bars. So you would both steer and propell with the hands and arms.

Of course, when the string ran out on the hub spool you would have to stop and rewind it. Or maybe they could have a mechanism they could engage that converted the reverse wind up motion also to forward propulsion so you get forward motion on the "up" and "down" turns?

and if you think about it, we have belt drives now. So if the ancients really wanted, they could have come up with a foot drive train without the need for a chain. It could have been just a leather belt or string with spherical pegs interspersed that fit into a wood cog.