I could mention a hundred things that make The Lord of the Rings one of the greatest cinematic achievements ever. From the music to the cinematography, to the fact that it was the biggest gamble in the history of cinema, or that it's probably the best adaptation of a book in cinema history. I don't have the time to do that, though, so I'll cut it short. You don't need to show Sauron do evil to know he is. On the contrary, I think it would severely damage the movie. I won't get into details about Tolkien and his themes, but what Sauron is meant to represent is a concept more than just a bad guy. He doesn't do much in the books as well. But every evil deed, every villain and every seed of corruption in the story stems from him. You don't see him embodied on screen a lot, but what you do see is the consequences of his actions. The One Ring is the embodiment of Sauron, and Sauron is the embodiment of Evil. One of the reasons for this is Tolkien's passion for epic and myth and, most of all, his veeeeeery strong catholic faith. I agree that the characters are not at all psychologically complex (with the one exception of Smeagol, who I think is amazingly complex), but that's kind of the point. The Lord of the Rings is more meant as a mythological story than a novel, and the movies manage to capture that so beautifully that I find it hard to see anything I would change about them. Apart from maybe a couple of Legolas antics.

In short: it's a story driven narrative meant to represent an insurmountable fight against Evil incarnate and the corruptive properties of absolute power. It's absolutely not character driven, as it was intended as part of a new English mythology rather than a classical Hollywood flick. The movies reflect that and I think (and apparently the Academy does as well) that the result is one of the best (if not the best) things ever done with a camera, but to each their own.