This seems to focus on the denuded aspect of such language but the perceived value of introducing such language shows we can't see ceremonial uses as totally empty. For example the phrase "under God" was added to the already existing pledge by the efforts of religious organizations like the Knights of Columbus and by President Eisenhower who was a newly minted Presbyterian at the time. Further, the legislative argument looked back to the Founding Fathers and specifically to Lincoln's Gettysburg Address and the appeals made to God there.
Civilian justice arguably requires an adequate functional concept of God as a practical postulate of statecraft and regime design. As many philosophers of religion have observed we may not have by reason alone a concept of God that gives a clear sense of it's necessary or sufficient conditions, and that in practice most actual religious language is about some unknown that occupies a certain role. (Creator, Redeemer, etc.)
For civic purposes, God is the objective ground of universal and perpetual equity and thus sets the standard of justice that the formation, enforcement, and adjucation of the Laws must comport to. God also grounds the possibility of legal reasoning as the means to secure justice and fairness as opposed to appeal to the stars or combat. Finally, God fulfills the role of final Judge of statesman and grounds the probationary climate of the practice of the state, holding the leaders accountable and making it rational for them to maintain the rule of law in difficult situations. Thus, statecraft is not left to brute nature, either by seeing society as a jungle or as the debris of natural processes. God also grounds the society as a humane economy.
Because God is taken to exist, the possibility of God as a religious object of worship must betaken with prima facie credibility by the state. However, the paucity of the concept of God postulated for civil society does not speak to what the true religion must be but only provide space for the free and peaceful exchange and expression of the variety of religions. The state, though it presupposes God does not do so as a religious object but as a political one.
So it's not surprising that the God postulated by the state is a via media between a reductionistic secularism and revealed religion, a God that does "some work" but is not "robustly meaningful". A God who is most always referred to in the third person in our civil ceremonies.
Martha Neusbaum remarked that ceremonial deism was odd because Deists (i. e. philosophical deism) believes in an impersonal cosmic principle and not the personal God supposed by our ceremonies. She is no doubt thinking of Aristotle who only supposed "God" to be only an ultimate final cause, thought thinking itself, and having no regard for what's going on outside of it, while everything else is attracted to it.
Such a concept would certainly by stable and answer to some of the civic functions. Such a God would not be creative in any supernatural sense but only directive and only inadvertently so with respect to the world. Such a God would neither be imminent nor transcendent but simply be "at the edge of reality".
But the Deists of the Founding period were not Aristotelians. Even though not Christians the still held that God was a moral law giver and therefore a personal agent. This thanks to the work done between Aristotle and that time in which God was seen as first in the order of efficient and formal cause as well as final. God is agent and not just thought. In fact, even thought thinking involved agency in the same sense as pure act as classical theism. Further, even if pure thought itself takes no regard for those that it influences, it would still ground a principle of karma that rewards justice and punishes injustice in a next life.
Philosophical Deism is an effort to maintain a tension between God as Moral Ground and God as involved in the affairs of humans. But pushing the latter makes God so remote as to fail to fulfill civic functions adequately. But pushing God as Moral Ground makes God a propeaduetic for a rational revealed theology. Bishop Butler famously illustrated this in his Analogy of Natural and Revealed Religion, arguing against the Deists that the grounds and methods of Deism by parity of reasoning also supported Christianity.
But while Deism as a philosophy is in tension, a Christian theism that recognizes different spheres of responsibility mitigates those tensions. By recognizing that religion, politics, and science are distinct spheres from each other and also from the Christian worldview, they can recognize that the same God is primary in all spheres, have space to attend to the possibility of revealed religion on the religious sphere, while preserving God's role as a political object in the civic sphere where the share life with non-Christians on equal terms.