Really great idea, integration to non-oauth email providers seems like a hurdle though. Requiring full credentials is a less good idea.
If they add post-read-filtering of the inbox it would handle some sorting after the work is finished as well. For example, filter all mail from X flagged with label done to that customers folder.
We've just added some google labels support, so we'll be able to handle this in the not to distant future. There isn't enough time in the day to code fast enough :)
Sounds really neat, I've solved it for my own personal needs with some imapfilter scripts, but the thought crossed my mind to make a service out of it.
The reason not to was the tricky part of needing customers imap credentials and store them safely.
Integrating to tightly with gmail would perhaps make it harder to support other imap platforms, since gmail can do more in some areas.
If they add post-read-filtering of the inbox it would handle some sorting after the work is finished as well. For example, filter all mail from X flagged with label done to that customers folder.