Always run the full algorithm for a given mime type before considering
fallback types.

This includes considering installed applications capable of handling a
particular mimetype, even if such an app is not explicitly marked as
default, and there is a default app for a less-specific type.

Specifically, this often helps with cases of installing apps that can
handle a particular subtype of text/plain.  We want to take those apps
in preference to a generic text editor, even if that editor is listed as
the default for text/plain and there is no default listed for the more
specific type.

Because of the more holistic approach taken by the algorithm, it is now
more complicated, but it also means that we can do more work while
holding the lock.  In turn, that lets us avoid duplicating some strings,
which is nice.

https://bugzilla.gnome.org/show_bug.cgi?id=744282