While an idea like this sounds appealing it seems to assume all emails you get are a bother (either because they're irrelevant or because they're asking you to do something).

I get useful emails all the time (or at least, emails that I should see) and quite a few of these are from automated systems which don't give a damn whether I'm there or not. If a third-party support desk sends me an automated "Ticket Closed" email for a call I've had open for months then I need to be aware of that, and deleting the email doesn't really help anyone (except them maybe).