Fu Manchu was never really one-dimensional. Even in his first appearence he's remarkably pragmatic and even kind of thoughtful about what he does. And yes, he did develop as a character significantly, probably because Sax Rohmer realized that he was far and away the most interesting thing about the book and people liked him a lot more than Nayland Smith.
And I don't mean to brush aside the unquestionably racist stuff that was in the stories and the character, or the negative impact it had as a anti-Chinese propaganda tool. Out of all groups of people depicted in pulp fiction I'd say the Chinese had it worst second only to the Japanese, and Fu Manchu's huge popularity is a big reason why.
But Fu himself was much more than a racially insensitive caricature, and I for one would love to see new stories with Fu that tap into the complications the character has. Like how his motivations can be easily interpreted as a pushback against American imperialism or the horrors of the Opium War, how his name seems to give him away as not even being a real Chinese but instead a stranger to even those he claims to fight for. His relationships with his daughter, or other heroes of the time period, and so on.
I mean, if people can enjoy Dracula as a character despite of his grotesquely racist background and source material, it shouldn't be impossible for the same to happen with Fu Manchu as long as said racially insensitive background is not ignored but instead part of the subtext or straight up text