Assuming the encryption works as advertised, these schemes still allow more de-anonymization than you'd think. If you can compute very specific set intersections you can then get the set size low enough to reveal personal information with 100% certainty. (The set of people that bought X at M and Y at N and saw ad A at P and ad B at Q on a particular day might contain one person)

I do not know of any work in homomorphic encryption that claims to provide privacy guarantees in the face of side-channels from other sources (in this case, the side-channels could be phone location information for brick-and-mortar purchases, or web browsing activity timestamps for online purchases). It's not surprising that such work is hard to find, since merely computing correct query results breaks most reasonably privacy guarantees.

Also, the field of homomorphic encryption encourages people to use horrible cipher-modes like code-book, so guilt-by-association makes me highly suspicious of stuff in that space: https://en.m.wikipedia.org/wiki/Block_cipher_mode_of_operati...

The closest thing I know of that is workable is differential privacy, which (in laymans terms) introduces enough noise to make sure that overly-specific queries return random (but representative) garbage. Apple says they use differential privacy to train models on data that's stored on iPhones without actually giving themselves access to that data. Cool stuff.