The pitot-static system tells you the local airspeed, which is crucial because you fly in the air around the aircraft, not with respect to the ground.
Some sort of calculated back up might be helpful, but if the temperature or wind changes, the old value won't be the right one to maintain.
A GPS would have told AF447 that they were plummeting to the ocean and their groundspeed was way down from its cruise value, but they should have recognized that already from the altimeter. If there were skeptical of the altimeter, then the GPS might have provided a reference to start to piece things together, not that things should ever have been in that state in the first place.