You can in theory. But you have to choose: Either your data is encrypted or it's searchable. So in practice you prefer searchability.

Someone will probably say something about homomorphic encryption. Not deployed AFAIK, and deploying will be difficult if you want to ensure that you can search your encrypted mail but others cannot.

You can just search on the client after decrypting.

Oh, sure. Just download all of the messages. The problem with that is that you run into users like me, who have 25 years of archived email, a mobile phone, and who travel abroad.

If you exclude attachments, how many GB is that?

And when it comes to a chat client, it's not a big deal to pick which chat to search, vastly reducing the amount of data you need.

Just for a backups sake you should do it.

Are you suggesting that if I want to search on a particular device, then that device should hold my backup storage? If so, then I disagree and have chosen to store my backups on a device that's behind two locked doors, hard to reach, well protected from being accidentally disturbed, and which doesn't run unrelated software.

One can never have enough backup storage. I fail to see where your problem is.

One can never?

Okay, suppose that I were to back up all of my mail to each client device; three at the moment, or four if you count my old phone, which is still running because I haven't gotten around to wiping it. Then I'd have five backup sites instead of one. But four of those are backed up to the fifth, so the size of the mail backup would be quintupled, the activity on the backup device would increase and with it the number of possible occasions for a failure, and the age of my oldest backup would be automatically decreased to compensate.