OH! I thought that was part of the syntactic argument. I didn't realize that was just the "data" for the question. Sorry about that

Ok--
It doesn't mean anything really. It's emphatic.
"Cats play, as they do, with yarn."
"Linguists study, as they do, languages."
"Complaining, as they do, teenagers can be hard to get along with."
"Having, as it does, three sides, a triangle is the simplest polygon."
It's a way of emphasizing the habitual/inherent nature of whatever activity is going on for the subject.