If the problem is that you have to change email addresses when you switch providers, I think the easiest solution is to use your own domain. Then you can use whatever mail client you prefer, and switch between them freely; you really own your email.
As far as using Twitter in place of email goes, there's the issue of both parties needing a Twitter account to communicate. If you opened it up so that I could send you a message on Twitter without having one myself, you'd probably end up with something like jdoe@twitter.com[1].
Comments
If the problem is that you have to change email addresses when you switch providers, I think the easiest solution is to use your own domain. Then you can use whatever mail client you prefer, and switch between them freely; you really own your email.
As far as using Twitter in place of email goes, there's the issue of both parties needing a Twitter account to communicate. If you opened it up so that I could send you a message on Twitter without having one myself, you'd probably end up with something like jdoe@twitter.com[1].
[1] Relevant xkcd comic: https://xkcd.com/927/