With ANR headsets you can hear the engine, prop and all other noises including the spinning gyros just fine. ANR only reduces the intensity of the noises and it reduces the highest decible noises more than the lower ones. It doesn't make them disapear. You really don't miss out on anything and what you do hear is clearer because your auditory senses aren't overloaded and your ears aren't ringing.
If you have never flown with ANR, give it a try and you may be in for a very pleasent surprise. There is a reason many pilots will pony up the $1000.