You see, back when web applications were hopelessly inferior to native applications, we would use a mail client that ran locally on our computer to read email. We could use that same program to check mail across many different email providers... so if we wanted to change email providers, well, we could do that without changing our UI.

These mail clients used standards like pop3 or IMAP to communicate with the mail server. with IMAP, you can check email from multiple devices/computers just as easily as you can with gmail (though, for some reason, most users were confused by IMAP; pop3 was popular long after IMAP was universally available. using pop3 from multiple computers is a huge pain.)

(Of course, this was back before everyone and their dog had their own domain name, so switching email providers was still a huge pain in the ass.)

My suggestion is that everyone pick an IMAP client to handle their email, then switch between providers as-needed.