Isn't the character type of the old Asian who is just there to teach the white guy mystical Asian secrets a racist concept to begin with?
Yep.
The Ancient One and the Mandarin were racist stereotypes to begin with. I kind of forgive that, in that they were created in the early 1960s, and I don't generally blame people for failing to be ahead of their times. But using them as-is would have been bad.
So, is it better to leave a racist concept like that or get rid of it by replacing it with a white person?
Well, as we consider this, we should remember that Ben Kingsley's birth name is Krishna Pandit Bhanji.
I think, with both of these characters, they are actually getting rid of the stereotype - Kingsley's Mandarin was nothing much like the original comic book stereotype one. I am expecting much the same for The Ancient One. They are merely reusing part of the concepts, and revamping them for the modern era.