It is a bad idea but not in the way you think. FHE hardware don't decrypt data on-chip. It's like using the Diffie-Hellman key exchange for general computation. The data and operations stay encrypted at any given moment while outside your client device.

The textbook example application of FHE is phone book search. The server "multiply" the whole phonebook database file with your encrypted query, and sends back the whole database file to you every time regardless of queries. When you decrypt the file with the key used to encrypt the query, the database is all corrupt and garbled except for the rows matching the query, thereby causing the search to have practically occurred. The only information that exists in the clear are query and the size of entire database.

Sounds fantastically energy-efficient, no? That's the problem with FHE, not risks of backdooring.