The first is that the premise is trivial at best and detrimental at worst. Let's evaluate my current mail situation. Considering that I'm a junior in high school, I receive a substantial amount of college-related junk mail (between five and ten pieces per day), in addition to perhaps three other items. I don't understand why it is preferable to sort though my mail digitally. It is already trivial to remove my mail, quickly glance through it, and toss the junk mail into the recycling bin in my foyer. Being able to do this via a web interface means nothing to me, and certainly isn't worth my money. In fact, I would argue that this worsens my mail-reading experience inasmuch as it creates a level of abstraction that increases the time necessary to deliver the mail I do care about, like The New Yorker. Our mail system is already a travesty. I'm certainly not going to do anything to make it even slower.
The second issue, as mentioned before, is trust. I simply don't trust a startup to open my mail.