I'd put the chances near unity that this would become heavily policitized pretty much immediately, if implemented, and that this would taint any of its results. At least in the United States, policymakers stand to gain much more from flattering their respective religious bases (substantially different between parties, but theistic either way and not heavily invested in a rational approach to religion) than from publicizing potentially uncomfortable findings.
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.)