Assuming you could teach furniture carpenters to build bicycles from wood, the advantage bicycles could provide over horses would be cost. You don't have to feed it, and it would be cheaper initially too I am sure, at least after it gets to the point that you don't have to specially commission them. Horses used to be very expensive.
Over cheaper modes of animal transportation (donkey and cow?), the bike would probably have a speed advantage.
The real downside in both cases is cargo capacity and manual effort, but I think it could have made sense as a poor-mans transport. Barring that though, it could just be a toy for the rich.
At ~5 MPH for most travel, bikes wouldn't have had significant handling problems, though shock absorption on cobblestones would have been a bigger issue.
If you ever get up close and personal with a stage coach, you'll likely find the suspension interesting, though most in my experience are spring-loaded but undamped. There are reasons long land journeys weren't generally looked forward to.
I was thinking in terms of comparing riding a horse to riding a bicycle. I'm sure stagecoaches were miserable rides, but I don't think any kind of bicycle based technology would really compete with one.
Over cheaper modes of animal transportation (donkey and cow?), the bike would probably have a speed advantage.
The real downside in both cases is cargo capacity and manual effort, but I think it could have made sense as a poor-mans transport. Barring that though, it could just be a toy for the rich.