In a universe as diverse as Star Trek (infinite diversity in infinite combinations) that isn't a big problem. Especially considering Ricardo Montalban is not a dark skinned North Indian Sikh. There are plenty of racially diverse characters with mixed heritage, but in the original series they were more separate to drive the point home that they have zero problems working together as different races.
The character Khan is a genetically engineered human named for the man who helped create him, Noonien Singh (Also the creator of Data in TNG). Essentially he does not have a country of origin or cultural bloodline. He is the engineered product of a group of scientists who identifies himself as not even being human. Other augments (as they are called) like him were bred in incubators, effectively having no parents.
The REAL flare up over a white guy playing Khan, as even George Takeai said, was that Khan was supposed to be the pinnacle of human evolution and the superior being, so making that a white guy was offensive. However that completely misses the point of Khan. Khan represents the worst of the "old world" and its ideologies including racial superiority (Khan wants to purify the galaxy of species inferior to his own, including natural humans). But when a diverse group of people from a unified society where race is no longer an issue can defeat him, and by extension his ideals, it strikes a powerful blow. The race of khan should not be an issue considering what he is meant to represent as a literary device, but in my opinion him being white only drives the point home more bluntly that en masse white people are the ones that have held Khan's ideas, especially in the society which created Star Trek. Defeating a "master race" type villain really drives the point home on the real life social issues the story is commenting on. The white guy may be labeled as superior, but he's also the villain who is defeated because he wasn't as superior as he thought he was.
In an industry guilty of making people of color more commonly the villains, which has been another social issue, making this villain a white guy who is brought down because he WASN'T racially superior is ok in my book.