Maybe they too are fake/non-existant, a kind of honey pot to catch external spy agencies or a way to distract them.
Apparently they forgot how to make it, and had to reverse-engineer the original process. It's a neat story. See page 20 in https://www.lanl.gov/science/weapons_journal/wj_pubs/17nwj2_...
Aerogels, having been invented in 1931, weren't directly secret, given that they appeared in scientific publications.
Meanwhile, it seems that the existing physics package designs had been optimized for a specific type of aerogel, and that the level of precision to maximize yield accounted for impurities introduced by the specific technique that had produced previous batches.
The nature of the secret seems to have been a gremlin, both incidental and intrinsic to the design of the contraption. And yet not aerogel specifically.