There's no option for OPTING OUT of the REGISTRATION database. The best you can do is register the airplane in the name of a corporate (or similar) entity that is not so easy to look up on line.
The Federal PRIVACY ACT puts strong limitations on the databases that the government keeps on PERSONS (as opposed to aircraft which is property). The FAA caught wind that they were violating this by making the airman data public back in the 1990's. They got a limited exemption as part of AIR-21 which requires them to send you that letter allowing you to opt out.
The problem is, that they MISREPRESENT that opt out. The LAW says that you opt out of "ALL DATA" where the FAA blocks only your address. Further, my contention is that the exemption only applies to pilot certificates and ratings. The stuff on your medical should not be there whether you opt out or not.
The disingenous LIARS at the IG and the registry feed all sorts of bull about his information to include:
1. Legislative history shows the intent was purely for the postal address. This is bunk. In fact, I dug through the several weeks of history on AIR-21 (the FAA appropriations bill this was part of). There's no "history" in the proposal other than it's mere insertion into the legislation.
2. It is necessary for employers of pilots to get that information. Also bunk. The FAA has a different, not PUBLICALLY, accessible system for vetting airline employees. The airlines should not be using the public airman registry. It's not the one that's been vetted for that purpose.
It was easier getting NIGHT fixed.
