rok, I suspect we all appreciate that labels can be difficult.
I readily admit that the Swingle Singers are not jazz, not in the strict sense. Is "jazzy" a better term? So far as my limited musical knowledge takes me, they are singing Bach's score. But it is not only the tempo, rhythm section, and their choice of vocal blends; the contra punctual form of Bach's music matches the lines followed by instrumentalists in many jazz groups.
I've told this story before, although maybe not here. In college I got together Wednesday evenings with friends after studying to share recently acquired jazz recordings. One evening after an hour or so we were at a loss for what to listen to next. The guy who's apartment we were visiting put on one of the Brandenburg Concertos, I'm no longer sure which. Up to that point I'd had almost zero exposure to classical music, but I loved that! So I began buying recordings of Bach and other Baroque composers. Only later did my classical music tastes expand to other periods. So maybe that is why I might accept Bach in the jazz realm more easily than you.