As a classical pianist, I am with Glen and John on this one. Damper whoosh is as much part of the piano timbre/sound as key-clicks are part of the sound of clarinette. We have become so accustomed to the sterile sampled piano sound that we find those noises objectionable. And we shouldn't be.
Personally, when playing I have at times intentionally induced some damper whoosh (quickly releasing the sustain pedal and depressing it again) to put the strings into vibration which then adds to the overall richness of the notes coming after it, by increasing sympathetic vibrations.
Also, as John said, most listeners will not pay attention to it, as the human ear has a tendency to filter out "unwanted" noises subconciously. You will notice them because you are too close to it (i.e. you are the performer, the recorder and you are too aware of all the warts, noises, performance issues, etc). The listener is more objective in that regard, so they will just listen to the music.
If you find the noises so objectionable, you can pretend to be Glenn Gould and hum over your playing
No engineer ever removed his hums from these recordings and personally I think they add to the charm of his performances.