Supplementing GTD workflows via email integration is more compelling than trying to replace people's existing desktop or webmail clients.
A list of integration scenarios that are more restricted than full integration is available on the above proposal.
Running a "normal", open-to-the-world generic email service has issues to build in-house (spam, overhead); outsourcing is a real option. But there may be various interesting ways to run limited email integration services without running a standard (SMTP+POP+IMAP+SPAM) email service.