Sometimes.... it's a fine line between costume and uniform. The original bicycles were created in small shops by skilled metal workers. The cap we call a "cycling cap".... was first known as a wielders cap. I think the small brim/bill would be useful for deflecting falling sparks or splatter... without deflecting grinding debris into the eyes. Of course shop owners would buy hats, gloves, even shirts in volume and have their logo's printed on them. I would guess some of the "branded" materials were given to customers as swag. Hence the functionality of the cycling cap.