From much more experience doing this than I ever imagine I'd have, having to enter each and every purchase manually, deal with alerts, and then manually go back and forth with customer service is painful. There's more work to do on our end (we're early stage). But extreme user simplicity is what we're going for.

I really like the idea of the service, though. It just doesn't work for me as it is right now...

Until you save one person a couple of thousands on a large purchase. I wonder if they could do this with car purchases or similar high-ticket items.

If the user would forward things to this account (either manually or automatically) that would work, but few, if any, places make a distinction between your ability to login and the identity you use to receive email, so you couldn't divorce the authentication from the notification. This account would also get password recovery emails, for example, and potentially expose more than just the receipt emails. It would partition the more personal stuff, but is only a partial solution.