I can only say that I have only seen a postman deliver mail in a bag on a bicycle on British TV shows. In the US, mailmen drive and they don't deliver mail from a bag.
They walk door to door and deliver mail from a bag in many older urban areas, small and large, in the United States.
In my town, we have both, and the same person may do both. I've seen my delivery woman park her truck, then get out and walk, delivering from her bag. She walks up one side of the street for a couple blocks, then back down the other side, returning to her truck. She'll then move her truck up a few blocks.
However, that being said, there is a movement to go to truck delivery only. I kept getting "friendly' reminders to install a curb side box. I even got a 'final notice', about two years ago. I bought one, but never got around to installing it, and since they keep delivering my mail, I haven't had much incentive to do it.
Apparently if you don't have a curbside box, they still have to deliver to your house if you have incoming mail. However, if you don't have incoming mail, they're not obligated to check your box for outgoing mail.
Since I always drop any outgoing mail off at the post office anyway, it's a non-issue for me. Maybe this summer I'll finally get around to installing it, since I already bought the silly thing.