Maybe I'm wrong on the typical path of forces and cameras may employ different routes for actuating advance sprocket and take-up spool. (I know of different linkage to the shutter cocking and for film rewinding.)
The advance sprocket must rotate for a fixed amount of "unrollment". The take-up spool must yield the same unrollment. As its diameter increases there must be a way to decrease the degree of rotation of the take-up spool. A friction device just before the take-up spool is the most simple solution, and it comes up for any tolerence in mechanics and film thickness.