A situation in which an embedded system communicates to public
outside systems via HTTP & HTML would seem a logical candidate
for an embedded, custom, server.  It might call internal
routines instead of using the CGI mechanism, for instance.
It might be reasonable to skip support of some HTTP things,
in which case, as has been demonstrated, it could be quite
small and easy to create.  It could be carefully vetted,
which might be rather more tedious with a full-up COTS server.
  I find it more difficult to imagine scenarios where a
custom browser of limited capability would be as worthwhile.