This includes encrypting unencrypted messages, after arrival, to the user's own key(s).
Keep in mind that data security consists of numerous elements:
* Access.
* Encryption-in-flight.
* Encryption-at-rest.
* Encryption-to-client.
* Rates of transfer or exfiltration. This has numerous elements, one of which I refer to as the Pennyworth limit. Bruce Wayne has a butler who is privy to Wayne's innermost secrets. But he is not also privy to the secrets of all of Gotham. And even at maximum data transmission rate, can only reveal a few hundred words a minute of information, on the order of a megabyte or so per day (in textual form). Our electronic devices and systems have far more information, in far more detail, on far more people, available at far higher rates, and for the most part, can be tapped with absolutely no awareness by the subject of the information revealed.
* An "entire mailbox" is only a few hundreds of MB or GB, and arrives incrementally. Since reading information client-side requires transmission of that data in any regard, the information is already client-side. There are now microSD devices with capacity approaching or exceeding 1 TB and above. Storage itself is not a limitation.
Unencrypted email can be indexed, the indexes encrypted, and transmitted to the client device.
I'm not saying that the technical considerations here are simple, but the costs of failing to address them, quite literally the future of liberal democracy, are too high to not do so.