Assuming by "email server" you mean an SMTP server, that's not quite what I'm talking about. What I'm looking for is more akin to a mail store, an IMAP server, and functionality for sending and receiving via actual SMTP server(s). That's a separate set of responsibilities to reading or composing messages, which would just need something like an IMAP-compatible client. And all of those are separate responsibilities to a full SMTP server with all the administrative overheads that comes with.
You could do the back office part of what I'm talking about with tools like Dovecot and fetchmail today, running on a Linux server with a standard Maildir backing store, and then you could run Thunderbird purely for its IMAP client capabilities as a front-end, but setting it all up and maintaining it is a horrendous hassle and it's far too technically demanding for most users anyway.
However, separating the receive/store/send responsibilities from the read/write responsibilities would mean you could have a single store on any server with relatively straightforward set-up to actually send and receive mail, which you could then access from any device on your network (or over a VPN or the Internet if it's running on a remotely accessible box), without the risks and overheads of both running your own SMTP server and trying to get mail from it actually accepted anywhere else these days.
Now that everyone's got PCs, laptops, tablets, phones, and who knows what else, I think this is probably the single most limiting problem with traditional offline email clients compared to all the hosted webmail services, but it shouldn't be necessary to incur all the downsides of those hosted webmail services just to access your mail from different devices.
