As I understand it the Right was not against Civil Rights per say.You're talking about only one element of the right. Don't forget that a great many "conservatives," in the South and elsewhere were conciously trying to "keep them in their place."
I would argue that these people were not truly conservative of the real America, but they would have given you a really good argument at the time that they were. And, unfortunately, very few true conservatives were willing to go after these people, since they needed their political support to fight Communism.
An understandable point of view, but sadly mistaken, IMHO>
I agree that there were those that would use the debates to further a racist agenda.
But too many on the Right are given the label and so the debate is tainted. Really the entirety of the Right is considered disqualified because the actions or thoughts of a few.
In reality many democrats were involved in the same agenda or worse and yet most do not disqualify them from the debate.
This is a case of forced perception.