If you prefer ice cream to cake when they are the only two alternatives, then why would you prefer cake to ice cream when a third, inferior, alternative is included?

You're thinking of a different meaning of "independence". A violation of the independence axiom of VNM would look more like this:

• What would you like for dessert, sir? Ice cream or cake?
• Ice cream.
• Oh sorry, I forgot! There is a 50% chance that we are out of both ice cream and cake (I know we have either both or neither). But I'll go check, and if we're not out of dessert, I'll get you your ice cream.
• Oh, in that case I'll have cake instead.