I don't want GBs of emails on my phone, desktop, laptop, wife's laptop, work PC, and anywhere else that I want to use email.

And I don't want to waste power and time syncing all that data between those devices just so i can do the same things i can now but less efficient (and with still more drawbacks)

No, it doesn't. This issue is not technology to store and search data from the client. The point is not wanting to download everything to every client in the first place.

Personally, I am a fan of p2p applications and doing things locally. Unfortunately, there are a lot of unsolved issues, especially when it comes to data storage, like you mentioned.