In ten years, if RFID passports are standard and readers are everywhere, passport covers may regain their usefulness, as barcodes will no longer need to be swiped through machines, and customs and immigration may well no longer require that passports be removed from covers. It's quite possible, actually, that they simply won't need to open the passports at all, but I expect it will take quite a few decades for them to realize this.
At the moment, however, passport covers aren't terribly useful. I've kept my recent RFID passport in my inside pocket, and haven't found that it gets terribly damaged. While it's embarrassingly nationalistic, and I do have a tendency to hide it from people (a passport is supposed to be an identification document, not a propaganda pamphlet!), I wouldn't worry about people identifying it. Passports aren't shown to many people, so if you're not identifiable as American, your passport isn't going to make it too obvious.
Besides, people who dislike Americans usually dislike typical Americans, and probably aren't going to change their opinion of you just because you have an American passport. Having an American passport doesn't make you that sort of "American."