Due to the inertia on large(r) aircraft this yaw rate would not be noticeable, so I suspect this is not a good option and no real alternative.
As many have pointed out the pitot static system has its limitations but it is relatively well understood. Problems can arise in parts of the system and cause erroneous information but this can be overcome by flying technique.
The problem remains how to be made aware of such failures.
This is an area that needs attention and development, as a recently released image of a heated pitot tube shows, where the tip was actually not heated due to design, where Boeing has since advised of a retrofit by switching polarity of the connectors.
Rather then looking for alternatives, efforts should be made to increase feedback to pilots when a system may have failed, as this is not always annunciated until a feedback loop records a complete failure. Such information would be useful to crew so to know when to ignore the affected instruments and use flying techniques to overcome a downgraded system indication.
We have all been told to fly the magenta line and to trust our instruments at some point in training, as well as to ignore those flight directors and instruments under certain conditions.
It is imperative that recognition of these conditions occurs. It may not always be easy when flying highly automated aircraft to notice such conditions as a crew. A feedback loop that would indicate a problem before failure would enhance the awareness of the crew to an impending failure or equipment downgrade.