In reality, this is a political thing and it's the boss's attitude. Not reliable is almost certainly just an excuse.

Bikes are more reliable than cars, absolutely. Especially for 2 mile trips, they're faster too. But the real issue is that your boss has decided he doesn't like it, and that's probably not going to change. Personally I'd just leave my car at the starbucks and drive it to the bank as needed.

Getting robbed might be an issue in many towns, I don't know. Personally I used to walk to the bank with thousands of dollars in the bank to do the deposit when I was 17 years old. I wasn't even disguising it, I had a vinyl zip-up folder with the bank's logo printed on it. But I lived in a small town.