The idea of a denomination is a brilliant achievement of American society. It is a distinctively modern idea - that varieties of the same religion, and even different religions, can freely compete within society.
There are two problems with this competition.
One is that it is not really a neutral standard, but really can only be fully embraced by people who do not think the differences between denominations matter. In other words, the denominational theory of religion works best for people who don't really believe in any denomination's religion.
The other problem is that the doctrine of denominational competition is not enough of a religious view to hold society together.
Modern societies have developed another distinctive religious view: civil religion. Civil religion was, originally, a religion of the (anti-Christian) state. The idea has broadened to mean the shared faith and symbols of the nation. Still, civil religion is most coherent as a cult that venerates the nation through the state, through patriotic myths, practices, and doctrines.
I had not fully appreciated before how much the ideas of denominationalism and civil religion go together. Really, they are two sides of the same coin. Denominationalism allows the old religions to live together in mutual toleration, if not respect. The price they pay is that each must accept an equal place within the new religion of the nation and its state.
This is an ingenious solution to the problem of religion in modern society. But it does require important modifications to all pre-modern faiths.