As far as I can see, the structure of supplementary constructions like
Karen, being ill, was unable to go
John – her father – was unable to walk her down the aisle
(i.e. compound nouns where the meaning is built up more by addition than modification)
don’t fit very neatly into either dependency or phrase structure theories, because they are not made up of a head plus dependents.
Given that these constructions basically function as appendages, you might say it's appropriate for the framework that describes them to be a theoretical appendage. Still, I was wondering whether there are other theoretical approaches out there that give more attention to this kind of construction, or integrate it better.
Also, is it fair to say that the more pragmatic the language, the greater the use of supplementation?