Do airline pilots use speed stability to cope with unreliable airspeed indications? If not, why not? There have been a couple of airline accidents in which unreliable airspeed indications (or the fear thereof) have played a role:


Birgenair Flight 301 in 1996, where one of the pitot tubes was blocked, causing unreliable airspeed indications. At one point, the captain gave full throttle at low airspeed and high angle of attack, causing the left engine to flame out, and producing a spin.
Aeroperú Flight 603 in 1996, where the static ports were blocked, causing unreliable airspeed and altitude indications. The altimeter read high, and the plane crashed into the ocean.
Air France Flight 447 in 2009, where the pitot tubes iced over, causing unreliable airspeed indications, for about one minute. The aircraft entered a stall and the pilots failed to recover from it. Presumably, the pilots were uncertain about whether or not the airspeed indications were reliable, which contributed to their failure to recover.


Now, correct me if I'm wrong, but if your airspeed indicator fails in, say, a Cessna 172, you can use the aircraft's speed stability to maintain a safe airspeed. Do something like:


Move the trim wheel to the takeoff position.
Keep the nose from rising or falling suddenly (unless you're near a stall), but allow it to rise and fall slowly.
If the pitch stays constant with no elevator input, you're flying at a reasonable speed. If it tends to rise, you're flying faster than that, and if it tends to fall, you're flying slower than that.


Once you get the nose to stay still, you don't have to touch the elevator much unless you change power or configuration.

Yet, I've never heard of airline pilots doing anything like this. Do they do this sometimes, and if not, is there a reason why they don't?

(For that matter, is there a reason why my suggested procedure isn't a good idea in GA aircraft either?)
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?