You mistake the conservative position on "states' rights".
My own very conservative position is that everyone must have his rights observed, but that these issues needed to be resolved through the political process. The political process involves everyone and consults everyone. Judges don't; this was particularly true of the liberal Warren Court and Burger Court Justices.
The problem with the civil rights movement is that the entire transaction was an exercise in grand-strategic coercion. The confrontational approach of the NAACP in particular, in contrast to e.g. the Urban League, revolved around a strategy of achieving total victory before a liberal, or at least a Northern, Supreme Court. Stop and think about it: the Civil Rights Acts of 1957, 1964, and 1965 were the sum and total of legislation on the subject during the days when these questions were being deliberated, and all the real movement was supplied either by Supreme Court decisions or by Lyndon Johnson's political coercion and threats of imprisonment.
That isn't dialogue, debate, or anything resembling a democratic process. It isn't anything but a steamrolling job, and as a Jacksonian democrat, i.e. a 20th-century conservative, I can't support the way that the movement's gains were achieved. It was a triumph in decree law -- court decisions and executive orders -- over a prostrate region of the country by the other regions, which were provoked by liberal media and liberal court decrees, all of it justified by polemic against white Southerners as a group, which is ironic given the nature of the original complaint.
The early history of desegregation showed some movement until the liberals typically overreached, assembling narrow consensus among the amenable, and trying to exclude "nonprogressive elements" -- shutting out people they thought wouldn't go along. That was what provoked the negative reactions of many white southerners, that and the segregationist rhetoric about Communists, appeals to racist stereotypes and race politics, etc.; but rather than engage the segregationists in debate, and instead of campaigning at the local and state level of politics, liberals went to court every time they encountered resistance -- every time. The result has been a tissue, a mass, of antidemocratic decree law which was promulgated precisely because of who the white Southerners were, a process of social change that, when it was applied in Northern communities later on, people up there found no more palatable, but which at the time, in the 50's and 60's, was felt to be "good enough" for the likes of Southerners, precisely because the process was intended to devalue them completely.
In short, there was no justice, but only a series of rigid decrees that put Southern whites firmly in the back of the bus. That isn't freedom, it isn't first-class citizenship, it isn't anything that would have been recognizable to the Framers. The Framers, I think, would have regarded the civil rights movement as a procession of horrors, fed by the violence of people like the Klan, but equally by the determination of Northern liberals to discuss nothing with the "crackers", but only to impose their own will selfrighteously on their own devalued "others", for the benefit of Southern blacks who lost many of their own rights in the process of so much curtailment of the ability of the Southern states to govern themselves.
To this day, no Southern state can hold an election. They have to apply to the federal government, which then allows them to hold elections -- but only after the Civil Rights Division checks over the redistricting plans to make sure that they're advantageous to Southern blacks (and of course, by the operation of zero-sum arithmetic, disadvantageous to Southern whites, who are thus officially second-class citizens of second-class states, forever). Thus the argument about "states' rights" is really a complaint that they don't exist any more, immolated in the attainder of the civil rights movement.
This damage is blinked by self-satisfied liberals (who don't care about the American political system anyway -- their taste is for authoritarian government) and their black clients who never enjoyed the freedom that their white neighbors did anyway. The result is a general degradation of the public's idea of what rights are, and especially of their inviolable nature. Liberals have deliberately assisted this conceptual decay with their proliferation of non-right "rights" to entitlements, etc.
The conservative Supreme Court justices have recognized the damage and have moved to try to attempt some restoration of the Tenth Amendment, which was the very amendment on whose adoption the ratification and adoption of the Constitution itself once depended. But they have an awful lot of work to do, and unfortunately most of the damage is popular, conceptual, political, and moral, and not easily amenable to repair by the court system.
'By what authority am I kidnapped in the street, and brought here by these dogs?' asked Monks, looking from one to the other of the men who stood beside him.
'By mine,' replied Mr. Brownlow. 'Those persons are indemnified by me. If you complain of being deprived of your liberty--you had power and opportunity to retrieve it as you came along, but you deemed it advisable to remain quiet--I say again, throw yourself for protection on the law. I will appeal to the law too; but when you have gone too far to recede, do not sue to me for leniency, when the power will have passed into other hands; and do not say I plunged you down the gulf into which you rushed, yourself.'