Do airline pilots use speed stability to cope with unreliable airspeed indications? If not, why not?
I don't believe the procedure you've described is formally recommended. Instead there is the unreliable airspeed procedure. The exact parameters vary among aircraft types but the basic principle is that by setting a specific pitch and power combination, you're guaranteed a certain airspeed/performance.

These procedures have existed for many years but perhaps the biggest change to come from the AF447 crash is an increased emphasis on this in recurrent training. 

See: What is the correct crew procedure for an unreliable airspeed alert?