The total opposite of structural language theories (such as Saussure) are functional language theories. Here represented first and foremost by Michael Halliday. Halliday is the father of SFL, systemic functional linguistics, from which genre pedagogy (see for instance Pauline Gibbons) seeks its roots. SFL seeks to analyze language from both structure and words in order to establish three meta-functions namely ideational, interpersonal and the textual. I haven’t seen any interpreting research using SFL (absolutely not a guarantee that there isn’t any), but guts feeling says it should fit well. There are some translation research done with SFL as one analysis model though. Apparently the problem with SFL as tool (only as hearsay as I haven’t tested it) is that it is descriptive, but does not really lead you any further. A bit like DTS, Descriptive Translation Studies. DTS was founded by Gideon Toury and became a revolutionary change in direction in translation studies, which up until then was almost solely prescriptive.
I teach from a book by Jan Svennevig Språklig Samhandling and the post is only my interpretation of his book in particular and of systemic functional linguistics in general.