So why is there no widely adopted app that fakes the overlay from an instant messaging service with the email back-end? This idea is obviously not new, i have found 4 companies that tried it [0] [1], one of these companies is Microsoft [2].

Is there a problem i don't see? Is mail to slow? - i have tried it between Gmail and some other service and it seems fast enough.

There is even Push-IMAP (P-IMAP) for the push notification.

So is there a technical problem? Why is the instant messaging over email not a thing?

[0] https://www.spikenow.com/de/

[1] Hop and MailTime

[2] https://techcrunch.com/2015/07/22/microsoft-introduces-send-a-short-form-email-app-that-works-more-like-instant-messaging/