The reason that we don't want babies to use a pillow is related to their age. Once they are able to turn over and move around in the crib a pillow is OK. Before that the concern is that they could push their face into the pillow and be unable or perhaps even unaware that they needed to move it in order to breathe.
I agree that it would work best to use a small pillow or one that is on the flat side so that their head isn't propped up too high causing their neck to bend forwards in an uncomfortable position. To be sure they wouldn't use it then.
Whether or not a baby uses a pillow once they're mobile is a personal decision. My granddaughter, 8, and my grandson, 5, rarely actually sleep on their pillows. By the time they're asleep they've also scooted down towards the middle of their bed.
I don't know why most adults use pillows. It must be a cultural thing. Some cultures do not use pillows. I've also know other adults to not use a pillow.