2) I can understand why you would want to see each and every email. Although I think some people get so much email that they would happily place some of the burden on the sender in exchange for the time savings.
I don't have this problem as I just leave emails I'm not immediately taking action on in the inbox. But maybe an auto-responder is a better solution for those that have this problem. It could explain that you're busy and hint that the sender can add the word "URGENT" to the subject to grab your attention in emergencies. You could then, with a few simple rules, manage your inbox without giving an external service access to your inbox.
Anyway, as far as anyone is concerned my inbox is always swamped. It's usually true, but if it's not I might get away with reading a few extra articles on HN without anyone knowing my inbox is screaming for some action.
