I think you may be making a false assumption here. Just putting a "website" in the cloud doesn't make it scale. There may well be dozens of choke points which are design limitations.
Equally we're an on premise shop and we've scaled without any problem during the sudden up-turn in demand. Our design is a long way from perfect but it was good enough.
I'm afraid the cloud isn't a silver bullet and can cause more problems than it solves.