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.
