To do this you would have to download all emails on all devices all the time to index them. Kind of makes the whole point of a cloud based email moot, if everything is on my device anyway, and logging in and out resets it all - might as well use an email client app.

Without producing confirmation side channels and so on? Doesn’t seem that trivial- at least it needs more thought than assumption of simplicity

We do it in a fairly straightforward manner right now. Check out the blog I linked to above - generate an in-memory index, end-to-end encrypt it, and store it browser storage. It's only decrypted in memory.