Here is a patch that adds 'outline-search-function'.  However,
the requirements for this function arguments are quite non-standard:
its first argument should be a limit that is used in
outline-font-lock-keywords where MATCHER has the argument LIMIT.
Also the same function should be able to search in both directions:
forward and backward.  And the third requirement is that it should be
able also to be used as looking-at without moving point.  In this patch
the second argument HOW supports two values: 'backward' and 'looking-at'.
But this could be split to two separate boolean arguments,
this is not the final patch.  Also another optional argument
could be added to define an arbitrary property to search.

Then here the same function is used for apropos-mode and shortdoc: