Note that in the other frame (displaying *Messages*), the
mode-line-inactive face has inherited the underline from line-visited.
(If I select that window, then the mode-line face has not, that is just
in the current buffer.)  So remapping a face that has inherited faces
leads to side effects in other places...

Anyway, what I was thinking of is a really simple solution: Have
`face-remap-add-relative' loop over all children and remap them, too.
(I haven't actually attempted to write something like that, though.  😀)

-- 
(domestic pets only, the antidote for overdose, milk.)
   bloggy blog: http://lars.ingebrigtsen.no