This is only first order accurate and the failure comes from higher order effects.
In your model you need the religious leaders to be aware of the ruse but still be benevolent actors. This is problematic because it makes selecting leaders from within the religion difficult (especially at lower ranks). By waiting to tell them the game many followers are likely to revolt as they've been told their entire lives and core beliefs are a lie. That risks collapsing the system. By pulling leaders from outside the faith you garner distrust from followers (maybe this could work if those members can be said to directly come from heaven rather, but that's difficult to pull off). By selecting from the followers and not revealing the ruse your system has no self realigning mechanism. In all instances you suffer from the effects of a game of telephone ("Chinese whispers"), but probably the most in the last case.
In all situations you are still vulnerable to the rise of fanatics. True believers of any philosophy can be dangerous, and the more power they possess (such as leadership) the more danger they possess. With competing religions you justify division between true believers. Even if the leaders of competing are not, the true believers will create that division and organize. A small population is all that is needed in order to upset the system.
I think you're right that there is utility to many aspects of religion (not just Christianity) but holy wars are still bloody wars. It's important to realize that these features can be met through other beliefs. Some of the greatest features are community, as organized religion forces common ground and people that have differing opinions in disjoint domains (e.g. shared religion, different politics) are forced to interact, which leads to lower tribalism (though not sub-tribes).
Maybe religion is a local optima, but it would be naïve to discount its flaws. By its nature it is dividing, even if at the same time it is unifying. Local and global structures need not be identical.