I agree with the whole first half of this. I'd quibble with the notion that poor behavior for theists is only restrained by their beliefs.
It comes back to your very astute point that reasoning about ethics in moment to moment situations is very difficult. As a result everyone, theist or atheist, is operating off neural lookup tables for daily operation.
What matters in the end is "right action." What drives that action is only important, insofar, as it has a greater or lesser likelihood of misdirecting right action.
Intelligence itself is only a useful adaptation in that it gives the owner the ability to periodically update their lookup table when new conditions arise. Religion is not a restraint upon behavior, but rather a set of heuristics for updating behavioral responses.
Those heuristics are incredibly complicated because, as you noted, the game theory is incredibly deep. Thus instead of laying out an explicit logic, containing every possible branch, the truths evolve into stories that compress the information into a structure that is both easy to process and more importantly, easy to replicate between hosts.
This is super important, because social interaction is mind-bogglingly complex, lookup table updating can't just work for the smartest 1% or 50% it has to work for everyone. Story compression allows that to be accessible even to people baffled by basic logic. I'd argue that even the very smartest folks can't come close to processing the requiremed theory without extreme compression.
This is why I'd argue that theists are not necessarily better people, but a world with theists in it is infinity better than one without. It gives society a way to evolve and act cohesively.