Some data: Inria is (AFAIK) one of the 3 major public research institutions in France: CNRS (the largest, multiple scientific fields, 26k people), Inria (CS, 4k people), INSERM (Medical, 6k people). There are others, but they are much less staffed. Anecdotally, I worked in a CNRS lab and I can tell you that it was very much applied. But applied research was the stated goal of the lab, and I can't say about the rest of the CNRS.