Let's not forget that there are many aircraft - high performance jets, notably, that fly their approach using AOA as the primary reference.
If we had our choice I imagine that we would choose to fly solely AOA on approach, as it is the most appropriate measure of both aircraft performance and safety margin. Conventional aircraft are AOA stable, and as such we choose an airspeed that results in a given AOA - important point here...in a given configuration, at a given weight, the AOA is known, constant, and stable for any given airspeed (and vice-versa). The weakness that we have today is in the method of displaying AOA, and the smoothness of the data.
Previous posters are quite correct in that the AOA can vary with different wing configurations at a given airspeed; however, this can be easily compensated for procedurally. If standard procedure is to fly an approach with full flap at 9 degrees AOA on the guage one can easily accomplish this. Adjustments can be made for different configurations, either by aircraft systems or procedures. The important point is that the pilot has an appropriate target to fly to, and that he or she is capable of maintaining the target.
As an aside, when flying the Alpha Jet I would always cross reference my approach AOA to an airspeed once configured. Flying AOA was great until there was turbulence or gusty wind conditions, at which point the airspeed indicator would tend to jump around less than the AOA guage, and was therefore a better instrument to use.