Face: width, not front
Everyone sees a front surface; engineers measure the axial dimension of rotating parts.
To most people, a face is the front of something: the face of a building, a clock, a person. It is the side that meets your eye, the surface you see first. In everyday language, face means the visible or forward-facing side.
In mechanical engineering, face means something narrower and more precise. The face of a pulley is its width along the axis of rotation: the distance from one edge of the grooved rim to the other, measured parallel to the shaft. The face of a gear or sprocket is its length in the same axial direction. These measurements matter because they determine how much load the pulley or cog can bear, how far along the width a belt or chain can sit, and how evenly the contact pressure distributes across the teeth. An engineer ordering a pulley will specify its diameter and its face; the face is a critical dimension, not a description of appearance.
Other trades use face in ways that bridge this gap. In mining, the coalface is the surface of coal being actively extracted, the working surface where tools bite into the seam. In metalworking, to hardface is to deposit a tougher material onto a base metal by welding, building up a protective layer on the bearing surface. In machining, back-spotfacing creates a shallow recess from the reverse side of a workpiece. All of these senses keep face tied to work surfaces and working dimensions, not mere appearance.
The shift from face as front to face as axial width likely arose because the faces of a rotating part are indeed visible when you look at it head-on, but the term came to specify the actual physical span rather than just the visual impression. Engineers needed a term for the dimension along the axis, and face, borrowed from the visual sense, became the term for width. In construction, a facesheet is an outer covering, a use closer to the common meaning; in these contexts, the trade has preserved the visual sense while using it to mean a thin protective layer rather than merely the appearance.
The dictionary entry
face
(noun, mechanical engineering)
The width of a pulley, or the length of a cog from end to end.