Changes in pitch have the most immediate effect on flight conditions. This, I believe, is where some of the FAA's reasoning comes from.
On the ILS the most important thing is that you stay on or slightly above the glide-slope. If you go below the glide-slope you are no longer assured obstacle clearance. This is one reason the FAA recommends the pitch for glide-slope, power for airspeed method. If you ever find yourself below the glide-slope changes in pitch will most quickly get you back on the glide-slope. Using power will generate the desired effect but not as quickly.
When landing the approach should be set up well enough that you can use pitch for AS and power for altitude. If the approach is not working out, switching to the pitch for the aiming point and power for AS method will only increase the risk of stalls. When the approach is not working out the most common instinct is to focus on altitude and aiming point not airspeed. Using pitch to correct the aiming point puts you at risk of stalling, in this type of situation.
On short final both controls should be used to stabilize and finnish the approach. Only small calculated adjustments should be made at this point. If the pilot has made a good approach there should be little reason to make huge changes in airspeed and altitude. If they are much to high, low, fast, or slow they should go around. There is no reason to force the plane down.
I have always been taught that slow flight is used to simulate slow air speeds that will be encountered in the pattern environment. When approaching to land the plane will be close to stall speed and precise control of airspeed and altitude is necessary. It has been a while since I practiced slow flight but it would make sense that the same control method used for landing should be applied here as well.
I am not a CFI and therefor don't know what the best thing to teach in terms of the FAA is but I do know what makes logical sense.
I hope this helps.