The plane did have the air speed information, at least at the time it made the stall warnings. Starting at time 2h10m17s, the pitot tubes started working again, the air speed was available, and the Flight Director came back online [page 89 of BEA report]. That is why the plane was able to make the stall warning in the first place.
I also wonder if, once the FD was back online, could the pilots have just re-engaged the auto pilot and the plane would have fixed itself? This is just such a tragic accident when there was nothing at all wrong with the plane for most of the incident!