GPS speed is already available to the crew, although not in an easy to see place. It isn't designed to cross-check other instruments because it is showing entirely different information. The winds are changing the actual speed of the airplane across the ground (which GPS shows) and altitude affects the air density (which changes the airspeed shown on the airspeed indicator). The difference between the two can be vast, with no easy correlation. Altitude has similar issues, although not as severe.
Even if they could be "synced up", the problem comes in when there is a difference between the two. Which do you trust? GPS receivers can (and do) fail, and could show an incorrect speed as well. When the computer gets conflicting information like it did between the different systems in the AF447 accident, it gives control to the pilot to sort out.
From the time that the autopilot disconnected until the ice blocking the pitot system melted away, only 12 seconds passed. During that time the crew stalled the airplane and then a chain of events ultimately resulted in the crash of a perfectly functioning airplane. The exact same thing had happened numerous times before then without resulting in a crash.
After the fact, the NTSB identified numerous factors and made recommendations to improve crew training (which has been implemented) and hardware/software modifications to the airplane. Aviation very much takes a "continuous improvement" view of things, trying to learn from every accident and to try to prevent the same thing from happening again. Based on what I've seen, I doubt that this exact same scenario will play out again.