Here is another, not so new, idea:

Username is your public key, password is your private key. So we get end to end encryption and account ownership out of the box. Something similar to how .onion addresses work. Needing easy to remember addresses? Build aliases on top of that. Needing server-side automation? Handle trusted server your private key.

Also, almost everyone carry a 24/7 powered and Internet connected device in their pockets. The Internet - network that allows globally sending any data between devices. Maybe with full redesign third-party email servers should be made optional.

I believe we should focus less on how to redesign whole service and build more universal layers instead. First, vsending any data to any machine. Second, optionally remotely accessing our data. Third, mail-like format of data.