Ref T1806. Ref T5752. Currently, handleRequest() needs to return an AphrontResponse, but sometimes it's really convenient to return some other object, like a Dialog, and let that convert into a response elsewhere.
Formalize this and clean up some of the existing hacks for it so there's less custom/magical code in Phabricator-specific classes and more general code in Aphront classes.
More broadly, I want to clean up T5752 before pursuing T9132, since I'm generally happy with how SearchEngine works except for how it interacts with side navs / application menus. I want to fix that first so a new Editor (which will have a lot in common with SearchEngine in terms of how controllers interact with it) doesn't make the problem twice as bad.