Let's agree that the conservative movement was near-nascent at the time. It didn't really exist until the Goldwater campaign in 1964. The civil rights movement was actually much more mature than conservatism.
As you note, at the time, conservatism was more concerned with was national defense and a survival strategy for the Cold War. For all we know, civil rights may have been #3 on the priority list (after controlling spending) -- but #1 absorbed all of the movement's efforts.
Thus, it's difficult to call civil rights a conservative "failure". Especially when Republicans provided the majority of the congressional support for civil rights, while congressional Democrats were generally opposed.
Granted, the Republicans of the time can't be considered congruent with conservatism -- as the latter was still a nascent force (which is, basically, my point).
Conservatism developed later and apart from the civil rights movement. Conservatism, however, is intrinsicly more respectful of civil rights than is liberalism.
Important point. Also, certain churches that were still conservative at that time (the Catholic and Orthodox Churches) were very supportive of civil rights, and segregationists were even excommunicated. Of course, because these were not leftist organizations, they never got the credit they should have. But that's no surprise.
This is the key. I strongly believe that the way to the hearts and minds of black people today is through more exposure of conservative values (many of which they already embrace). Conservatism is NEW, and the best way for the black community to achieve any amount of progress today.