The role of public reason is not so much to eliminate or even diminish political disagreement, as it is to provide democratic citizens with reasons and arguments that, if valid and sound, they can accept as democratic citizens. Were laws and policies are decided for purely nonpublic reasons, it could not be said that democratic citizens are politically free. Their political power is being used against their will in ways they cannot endorse as citizens. Public reason then is a condition of political autonomy and collective self-rule.
The idea that there is no problem with surveillance as long as you have nothing to hide simply points to the complacency of the liberal view of freedom by contrast with the republican one. The liberal thinks that you are free so long as you are not coerced. The republican agrees, of course, that if you are coerced then you are not free. But freedom for the republican consists not in being free from coercion in respect of some action, but rather in being free from the possibility of coercion in respect of it.