It's actually very easy to call Sauron a two-dimensional villain because he is never seen in the flesh LotR. But I believe it really doesn't matter how one perceives Sauron because in closer analysis, he isn't the real "villain" of LotR. Evil itself is, especially as embodied in the One Ring. The struggles and hardships through which the protagonists must endure to reach their victory are more subtle than those faced when one faces a single, clearly identifiable foe. If LotR were constructed in the same way as most adventure stories, the "villain" can be defeated by taking out specific individuals, decimating their forces, destroying their armament. But Tolkien created a "villain" that didn't follow this formula. Yes, we see the forces of "good" planning their war against the enemy, the forces and allies of Sauron -- but even they know that victory in battle will not truly destroy their foe. The Ring at this point is more powerful than Sauron. It can live without him -- and even spread its evil without him -- but he cannot live without it. Sauron cannot be truly defeated so long as the Ring exists, and while the strength of evil is greatest when they are united, it is the Ring that is, so to speak, the linch pin of villainy in this story. It is the source of the greatest temptation and corruption -- in a way, it corrupts even Sauron, who allows his need for it to goad him into hasty and unwise actions and presumptions that are to his ultimate detriment.
So is Sauron a two-dimensional villain? Likely so, in my opinion. But is the Ring two-dimensional? I think as villains go, it is vastly more subtle and insidious.
As far as the "Tolkien is a racist" cant goes... That argument will probably go on forever. I don't believe he was, but I have learned that this is an argument one cannot win. People who are determined to see his writings as racist will stick to their guns no matter what I or anyone else says. It's that kind of an issue. For myself, I believe that if he had been truly racist at heart, he could never have written Sam's very compassionate reaction to his sight of the dead Haradrim. A bigot would have thought "there's no good Haradrim but a dead Haradrim," and rejoiced in seeing one dead -- and would have written that kind of a reaction into his story. He didn't. The orcs are the closest he comes to wholly vilifying a race (if they can be called that), and even they are not exempt from some expression of compassion from the author. "For me, I pity even his slaves." (Gandalf, at some point or another
).
Perhaps a "modernist" point of view requires all writings to conform to whatever is the politically correct mode of expression in current times, but if that is so, then I believe it does the unwise thing of throwing out the wisdom of past ages, expressed in its own language and in its own way. There is value in that as well as in modern ways of doing things, and something is lost if everything must be reinterpreted to conform to current standards and practices.
All just my humble opinion, as ever.