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I also love ap Ezreal.
But you explained it wrong. Lichbane IS amazing, but that adds physical damage not magic damage, and Ezreal doesn't do enough magic damage to be called an ap "carry".
His burst in teamfights is amazing but that's only when you hit both R and W, AND when you are stacking AP, NOT Lichbane. Lichbane gives you a lot of damage, but in teamfights you need all the AP you can get.