Richard Posner, appellate court judge and prominent intellectual, and Gary Becker, economist and Nobel prize winner, both posted about state sponsoring of religion here.
As is often the case, they both analyze the Establishment clause from an economic perspective.
Posner argues that religion can be subsidized if an equal subsidy is used to further secular interests. But he also agrees with Hume that subsidizing religion could lead to a decrease in overall religious practice and belief because the clergy would get satisfied and lethargic since their salaries are not based on performance and are constant. Moreover, the monopolistic nature of a state religion would decrease the variety, leading to less support for religion. For the most part Becker makes the same point, that competition allows religious movements to grow and government subsidies would create a monopoly, which would stunt the introduction of religious ideas into the public arena.
Posner ends with empirical evidence that shows that states with an established religion have less religious citizens.
As usual Posner's analysis focuses exclusively on the economic arena (economic defined very broadly), but ignores the moral issues. Posner's preeminent work has been on Law and Economics, which is the discipline of using the law to achieve the most efficient outcome, efficient defined based on greatest satisfaction for the greatest number. This is a simplistic analysis and for more see here.
Posner ignores the moral aspects of establishment. The two areas of societal structure are fairly distinct. For example, let's assume the government passed a law barring women from becoming lawyers. There are at least two ways of evaluating the law:
We could say the law is morally wrong. Perhaps it's unjust to bar a person from a profession solely based on gender because it violates the basic tenets of fairness and equality. This analysis ignores the efficiency of the law and is not subject to empirical proof either way.
We could also say the law is inefficient and therefore challenge its utility. Barring half the population would lead to the less efficient result of having less qualified lawyers. Allowing women to become lawyers would generate better lawyers because its logical to assume that some women are better than some men at law (after spending a year in law school I can personally testify to this fact). So its inefficient to deny women a legal profession.
This analysis is subject to empirical study. We could do research to determine if some women really are better than some men. If not, the law might be justified on those grounds.
Law and Economics only tries to shape society based on the second method. That's the primary flaw of Law and Economics. The moral issue cannot play an important role. Posner, though, will sometimes make arguments that try to include moral issues. For example, he probably would argue that denying women legal employment would decrease their happiness, therefore leading to a less efficient society given the diminished satisfaction of half the population. But in reality, while that argument does try to integrate morality into the equation, it cannot properly do so. Perhaps he can grant it more weight, by arguing that the diminished happiness is sufficient to outweigh the benefit of having better lawyers. But that's hard to really say. The moral question does not properly lend itself to equations.
Do not take this as an argument against balancing rights, which every government must do and every citizen should support. But unless someone can figure out a way to properly integrate moral questions into economic equations, Posner's arguments are lacking and always will.
Oh well never got around to the Establishment question. Maybe next time.