I'm not sure I agree with your thesis, but here are some ideas you could use. Basically, you want to look at what factors made the white population (particularly outside the South) predisposed to accept the arguments of the movement at this point in history. In other words, think about what would have happened if Martin Luther King Jr and all the other leaders had been around 50 years before they were. They could have had the same abilities and yet been unable to succeed. This would be because of a variety of factors:
- WWII. This was a war where the US was explicitly fighting against a racist enemy and was taking pride in its own democratic and tolerant system. This made Americans more likely to buy into the idea that rights should be extended to all.
- The Cold War. The US was competing with the USSR for the support of brown and black people all over the world. Keeping segregation made us look bad and made us less competitive in the Cold War.
- Movement of blacks to the North. With the migrations in WWI and WWII, more blacks were in the North where they could vote and were important to politicians.
Finally, you could argue that it was the nature of the movement's demands, and not the leaders, that made it work. After 1965, when MLK was pushing for things like more economic opportunities for blacks, he was not nearly as effective. The demands of the civil rights movement were relatiely easy to accept because they were for simple legal equality, not for anything that actually would have cost anything for (especially Northern) whites.