Don't forget most people bring their own personal biases into play. If they think riding a bicycle is exhausting for themselves, they may assume that eventually their employee will become too tired to bike to work.
You'll also have to overcome any biases against past employees. Perhaps they hired someone previously that had a DUI, was forced to ride a bicycle, and rode a POS that they didn't oil the chain or maintain.
Also bicycle mishaps are more injurious than car mishaps. Minor car problems don't cause injuries. A broken chain can throw you to the asphalt and skin a hole in the knee of your pants, and have you coming into work seeking the first aid kit.
If you live somewhere where traffic jams are common you'll have an easy time I think winning the reliability argument. Just wait for the co-workers to all come in late (stuck in a big backup) while you are on time
