I don't think the "modern state" is conscious of this as a desire to control the conscience. But I do think that this is effectively what happens in an allegedly free society when any group within it is uncomfortable with what another group does.
Look at Iran: their government oppresses the Baha'is. At least one Muslim scholar has suggested that this is because of where faith meets politics. The Iranian government is uneasy about Baha'is especially because they perceive that the Baha'is are obedient to government not because they're afraid of government but because they're faith requires it of them. Leiter is perilously close to making the same sort of argument against religion as a whole—it is a higher authority in the minds of adherents and when the law of faith conflicts with the law of the land, they seek exemption. If this were a Baha'i matter and the US government said "Baha'is can't wear ceremonial daggers (or ringstones or whatever), the House of Justice would almost undoubtedly require Baha'is to obey the law of the land. BUT, I'm going to go out on a limb here and suggest that even that obedience would not be good enough for Leiter and others like him because they, like the Iranian government, would be disturbed by the fact that the Bahai's were not obeying the law of the land at al; they were obeying their own religious leadership.
But I'd have to ask Leiter, how do we decide which civil rights (and whose) shall be protected and for what reason? I've had dialogues with anti-theists before in which I drilled down to the core of the argument and it is prejudice and bias AGAINST religion. The Iranian government's claim that Baha'is actually commit crimes of subversion is undermined by the fact that charges will be dropped if the Baha'i will only recant her faith. I've had anti-theists step into a similar paradox when they admit that they would allow freedom of dress, behavior, and speech as long as it is not motivated by religion.
One respected blogger went so far as to suggest that any idea that came out of religion should be simply discarded. Which led me to ask if that was not as irrational a response as the fundamentalist believer who refuses to accept any idea that comes from a secular (or other) source.
We are human. And there will always be those among us who consciously or unconsciously want to control what other people think, feel, believe, and do. What Leiter has proposed is anomalous—"individual rights" that are only protected if they arise from an approved source, which is not a real protection of individual rights at all.
This begs the question: what if the State proposes to do something that violates the secular conscience? Would Leiter suggest that it ceased being a state if it relented because that constituency requested a "special protection?"