"The key turn in the arguments of today’s neo-traditionalists is their assertion that the old traditionalists lost because their arguments did not dig deep enough; their arguments were too American and, hence, too liberal. More broadly, however, is their insistence of the recovery of the Church and the family as public, even political, institutions. The neo-traditionalists do not mean to open up Church and family to subordination to the civil government. In important ways it is just the opposite. Nonetheless, these institutions do not dwell in the domain of the private. Consider, for example, Adrian Vermeule’s identification of the Church as a political institution, even if one of ecclesiastical rather than civil government. (Here and here.)
Conservative neo-traditionalists naturally deploy their recognition of power relationships in the social very differently than Brown. Nonetheless, this is notable move among conservatives more used to singing paeans to “voluntary organizations,” among which churches and families were normally included. (Oddly, given the age-old problem liberalism had with thinking about children.)
Conservatives neo-traditionalists, however, did not learn of power relations in the social from Marx, let alone Foucault. Implicit in the notion of “subsidiarity” as understood in Catholic social teaching is not a hierarchy in which the state delegates or defers to non-state entities, but recognizes the significance of non-state social organizations, particularly the family and the Church."