I've never had any problems with the alignment system. There is some language in regards to the alignment of certain races that hasn't stood the test of time well. But to claim that the alignment system itself is inherently racist is absurd, and shows a misunderstanding of the game and its alignment system.
D&D was not designed to be an ethical debate. It started off as a simple game of good versus evil. The alignment system was designed to easily put various creatures into simple boxes. Much like a computer game, the players are the heroes, and they slay the bad guys, and collect the treasure. As the hobby grew, the dungeon crawls of old transformed into more meaningful stories. Nowadays we tend to think a bit more about morality and our stories become better and slightly more realistic. But it is perfectly fine to still run your D&D as a straight forward kick-in-the-door dungeon crawl, with simple bad guys. There is nothing wrong with that.
I've run adventures in which the players fought drow who were all pure evil. We didn't stop to have long debates about ethics. They were the baddies, and in D&D you kill the baddies. It's a game, none of it is real, so who cares? There is nothing inherently racist about running your game this way. Drow can still all be evil, if you want them to be. But obviously we think a little bit deeper about these sorts of things now, and the language about racial alignment needs to change.
I think though that not much would change about the game if we removed the alignment system completely. And I know some people have mentioned Planescape and Dragonlance. I'm not knowledgeable about any of these settings, but surely even settings that have traditionally involved alignment heavily, can rely on something else, without large changes to the setting and stories? Alignment is after all but a definition of good and evil, and not a game mechanic (anymore)?