That you started by mentioning government pronouncements on diet makes me think of the disastrously-wrong advice that the US government has given for largely political reasons, and primed me to think of a similar outcome for a National Institute of Theology. Besides, I don't think there are still interesting questions left to ask about religion; we already know the answers, it's just that people refuse to accept them because they're unpleasant.
Lots of charlatans spend lots of money advertising diets and health products that don't work. We have a National Institutes of Health that, among other things, funds studies that make more-objective pronouncements on diet and health.
Argue pro or con: It would be a good investment (payback would exceed opportunity cost) for the US government to create a National Institute of Theology, that would fund research into theological questions. For instance:
Everyone, religious and atheist, should be motivated to want the answers to these questions to become widely known.
(Yes, I know this would be politically impractical. Pointing that out is not interesting. What is interesting, is whether such a NIT would be beneficial, supposing it could be made.)