I will follow Posner and try to discuss the general principles concerning the State and religion rather than the details of these Ten Commandment cases. To me, the overriding reason why the State should not make any law respecting the establishment of religion is the case for competition and against monopoly. Competition allows for entry of producers, including new religious ideologies, such as scientology and bahaism, or new forms of atheism, that cater better to the preferences and needs of people, be they spiritual needs or materialistic ones. Monopolies restrict entry, and hence preclude the entrance of producers with new ideas, including religious ones.
Throughout history, religions have tried to use the State to give them a privileged and protected position, and in this regard have been no different than telephone companies and airlines that have used government power to keep out competition. This use of the State to foster particular religions is found in many Islamic societies that subsidize teachings and practices of Islam, the Israeli State that subsidizes Judaism, or some Christian nations that use taxes to pay the ministers' salaries. As Posner recognizes, many other groups also succeed in getting the State to support their activities, but two wrongs do not make a right. Governments should not support particular religions, or other groups that feed off the State.
Competition usually increases the demand for a product compared to monopoly. As Posner indicates, this is one of the arguments Hume made against State-supported religion. Adam Smith in the Wealth of Nations made a similar argument, and a quantitative study by Lawrence Iannoccone tested Smith's claim. He found some support for the conclusion that religions flourished more when competition among religions is greater. The US stands out in this regard, for it has several thousand "different" religions competing for members, and it is more religious than other wealthy countries. However, fundamentalist Islamic countries and Christian countries like Ireland and Poland do actively support a particular religion, and they also have relatively high participation in religious activities. So they are counterexamples to the Hume-Smith-Iannoccone thesis.
Moreover, as Posner indicates, large state subsidies to one particular religion could lead to greater demand for religion than in an unsubsidized competitive environment. That is why I believe the case for free competition among religions comes mainly from competition providing opportunities for new religious belief systems, including atheistic beliefs, to cater better to people‚Äôs desires. To repeat, the case for competition among religions is the same as the case for competition in other industries: to allow entrants with different ideas and points of view to compete for consumer time, money, and other support without any government favoritism.
I support government-financed school vouchers for various reasons that I have discussed elsewhere. That includes support for vouchers for religious denominated schools, as long as they are available on equal terms to all groups, including explicitly atheistic ones. The State does have the right to exert control over the curriculums of schools taking government aid in one form or another. This control can include sharp limits on how much time a school can spend teaching religious doctrines, including doctrines against organized religions.
I am not competent to discuss the legal aspects of the two Ten Commandment cases, but where does this analysis of the case for competition and a level playing field among religious doctrines come out on the issues raised by these cases? As Posner indicates, most of the Commandments deal with ethical issues that would be supported by all groups, and are not really controversial. Those that are sectarian--say to worship only one God--can be opposed on grounds that they support certain religions against atheistic groups, or even other religions. Consequently, I believe the principle of a level playing field argues against allowing religious displays, and many of the other displays on public property mentioned by Posner and in the Court's decisions. Still, these displays, including the Ten Commandments, are far more innocuous then the many laws that give monopoly powers to telecommunication companies, domestic airlines, farm products, and various other industries. So relative to the harm caused by laws that rig the playing field in these industries, the extensive agitation over the display of the Ten Commandments seems like a tempest in a teapot.