For those interested, the reason it’s not the same as a backdoor is that the result of the computation done on HE data is itself still encrypted and readable only by the original owner. So you can effectively offload the work of a certain analysis to a server that you don’t actually trust with your keys.

Do iPhones have a BYOK system for people to supply their own keypairs? Or is their OS open-source so that people can see how the keys are being handled? Because if not, it just sounds like all it takes to break this is for Apple's OS that it controls to ship the private keys that it generated up to its servers?