I hate to say it, but in this day and age, Mickey Rourke is the frontrunner to play Genghis Khan in a new movie. There's something to be said about Hollywood's incredibly strong desire to:
1. Cast roles meant for minorities with white actors, especially when it comes to movies that seek to tell stories of other ethnicities
2. Create a period piece in a typically non-white nation, but still center it around a white character to frame it nonetheless (Chinese political revolution? Let's focus on the white priest. Thailand is devastated by a tsunami? Forget focusing on the locals, let's make the white tourists our main characters, who can afford to seek refuge outside the country.)
3. Suppress job opportunities for minority actors in the first place. It's one thing to say that actors have to work hard in order to get roles, which is obviously true. But if minority actors don't get roles, they don't build experience nor an audience that would pay to watch them, as well. It's a vicious cycle that leads to Hollywood leading roles not matching up with American racial demographics. And as the Fast and the Furious movies have shown, people *will* pay to watch a diverse cast in leading roles, but Hollywood doesn't seem to get that memo.
As for the controversy about Khan, the internal logic isn't perfect by any means, but externally, it really helped Montalban's career, who in turn fought to increase visibility for Latino actors, so that the Latino acting pool wouldn't be relegated to just a handful of brown people for leading roles. Indeed, we should consider the fact that Roddenberry intently cast a non-white for the role of Khan despite studio heads, and that the experience of playing a complex character like Khan partially inspired Montalban to create the Nosotros Foundation. Thus, in its own way, Khan being played by a Latino (or the fact that a Latino actor got to really show off his acting chops) was a trailblazing move in its own right, and Montalban's performance is one very strong reason why Khan is consistently ranked among sci-fi's greatest villains -- it showed 1960s audiences that Latino actors could be more than mere henchmen or drunks.