setting a precision has no direct effect on coordinates of geometries, but merely set an attribute tag to an sfc object. The effect takes place in st_as_binary or, more precise, in the C++ function CPL_write_wkb, where simple feature geometries are being serialized to well-known-binary (WKB). This happens always when routines are called in GEOS library (geometrical operations or predicates), for writing geometries using st_write, write_sf or st_write_db, and (if present) for liblwgeom (st_make_valid); also aggregate and summarise by default union geometries, which calls a GEOS library function. Routines in these libraries receive rounded coordinates, and possibly return results based on them. st_as_binary contains an example of a roundtrip of sfc geometries through WKB, in order to see the rounding happening to R data.
The reason to support precision is that geometrical operations in GEOS or liblwgeom may work better at reduced precision. For writing data from R to external resources it is harder to think of a good reason to limiting precision.