I've attached a patch which implements a `browse-url-with-function'.
However I am not sure if we want to add more `browse-url-*' function
variants, since this will make the `browse-url' API increasingly
unclear.

An alternative could be to add keyword arguments to `browse-url' for
consolidation:

For example:
(browse-url url :kind 'external)    ;; replaces `browse-url-with-browser-kind'
(browse-url url :via some-function) ;; alternative to `browse-url-with-function'
(browse-url url :via secondary)

Daniel