Here's the problem I see:

- Almost nobody runs an email server on their mobile device or even tablet. They connect to an email server.
- Unfortunately, a third party server would have to either compromise anonymity, or serve every message to the phone (not feasible due to mobile network limits).
- Running your own server requires a server (not free, in contrast to most people's email), or having a home computer reliably on for a good portion of time at least every two days--more if you want to have email-like speed.
- Even if the home computer or server was being run, it's nontrivial to securely connect to it (required if the server isn't serving every message on the network to the client).