+1 for the point about Thunderbird. We need good open-source mail clients. I'm not sure we need a Firefox OS, and regardless they just don't have the market power to make it work.
Unfortunately Mozilla [ com | org ] have shackled themselves to Googlemail internally, so seem to have abandoned any aspirations to fight for openness in the domain of e-mail.
You might want to check out claws-mail. You get notification via a plugin (on debian/ubuntu, the plugin is in another package, but it's very easy to get).
Take a look at https://nylas.com/N1
> it works like a traditional mail client.
Except it's not just a client.