Earlier quoted context omitted.
The inverted index could simply be built/live on client side. The cloud is then just a backup data storage. Encrypted email is downloaded, unencrypted and indexed on your laptop/phone etc.
To be clear, you're proposing that users download their entire email storage to their phones and browsers before they can search or spam filter their email? And you think that this would be a compelling product for the average email user? I'm having a hard time figuring out if you're trolling.
Anyways, my point was not to suggest a full solution but to say that companies are not incentivized to solve this problem.