Just wish to point out that
To my mind, stopping noise energy getting to the inner ear has to be better than adding to it to cancel it out.
is not what an ANR headset does.
The headset earcup (each one) contains a miniature mike, which detects the instantaneous sound pressure level in the earcup, and there is a little amplifier (this is why you need a power source) which drives the speaker in antiphase, so as to cancel out the sound pressure.
So, it isn't a case of emitting extra sound to cancel out the external noise. The sound pressure level inside the earcup is lower, not higher.
Due to limitations caused chiefly by the chamber volume being quite large (the thing has to go over one's ear, after all) together with the speed of sound not being all that great, the cancellation works only at low frequencies, which is fine for attenuating piston engine noise.
The best ANR is achieved with in-ear systems, where you have a tiny air volume and a nice short distance to the eardrum... but then you don't really need ANR because you can mould a shape which jams nicely into the ear so external noise doesn't get in in the first place.
I gather a lot of aerobatic pilots use in-ear headsets, because standard headsets fall off. I was talking to one of these the other day; he was using the Clarity headset. I've tried the Lightspeed Mach 1 and got rid of it because it is useless without custom earplugs and I didn't want to mess around with it.