This may just be me, but if you use A to tune all the time wouldn't you become more accustomed to its pitch and therefore notice more often if it was sharp or flat?

Also, as someone who has been told they have perfect pitch (I haven't done any official tests so I'm not 100% sure), when I'm listening to music that may not be precisely on-key it doesn't bother me or sound "wrong", it just sounds different. That is, as long as the instruments are all tuned together; if it's just one instrument that's out of key from the rest of them, it does bother me. :P