There's no FAA requirement that a pilot use a TSO approved headset unless that pilot is replacing an "installed" headset (which was TSO'd) that is part of the type certification equipment for the aircraft.
There is no such thing as an ANR only headset. All ANR headsets provide passive attenuation, also.
ANR most certainly will protect hearing. Any method that attenuates noise to the ears contributes to protecting hearing, whether it is passive or active noise attenuation. Other industries that have noise damaging environments are now starting to use ANR headsets, e.g., engine rooms, industrial manufacturing.
If you want the proof, buy a Bose QC15 on a 30 day trial, and wear it as much as possible during a trip. You'll notice a dramatic decrease in tinnitus and an improvement in hearing acuity. Then, put back on whatever headset you were using. You'll be amazed. If you're not, send it back to Bose for a refund.
Mike