I wish I understood the actual data Alice needed processed to wrap my head around the problem space. Reading examples of what it could be used for leave me more confused.

Without knowing the actual pointed problem space the generalities of the schema just sounds like a snake game to me.

I can't get past where the veil is dropped for bob to take some ciphertext and operate on it in any meaningful way for him to create some machine that simultaneously can process encrypted data while being completely blinded to what the data is or the insights learned. Any form of ability to process a datum while simultaneously being blinded to it is logically incompatible to me.

Now, could we obfuscated it and lower the chances of an attacker being able to do anything useful with it or knowing what is being processed? Yes...but why muddy the definition of encryption in order to do this?

One example I can think of: privacy preserving ML models. Company wants to keep their model safe and also perhaps not to force you to run some classification on your phone for battery savings. You want your inference features to be private, because they e.g. contain your location data. You send them encrypted, you get encrypted results back and decrypt them.

It could have been obfuscated, but assuming we have HmE, obfuscation is more tricky to be done right.

Do we change the definition of encryption meaningfully in this process, though? If so, I don't see how really. It's just that out of set of potential encryption algorithms, for this purpose we would pick ones that are fit for it. It wouldn't be AES I guess... But it would still be encryption. Maybe a weaker one, but that weakness could be coincidental. I'm not familiar with any theorems in that space.

EDIT: I see a note that we lose ability to verify integrity of the data with HmE, but I guess it's still not really changing the definitions - just most of encryption used today also provides integrity guarantees as a kinda nice and desirable side effect, but it doesn't change that the encryption that doesn't still is encryption.