Make the scan and the visit routines look more alike.

This code shares a common origin but the implementations eventually
diverged.  Alas, a forthcoming change may cause the scan routines to
permanently diverge from the visitor.  Until that happens, we should
try to keep these routines as similar as possible.

Change-Id: Icf95b3bb4db704c8c7b4ab5aab84d9e091675bab