I think people should refrain from thinking they can tell what I clearly want from what I say, as their error rate is high. You assume that I want an institute that will reach a foreordained conclusion, rather than one that will try to find correct answers. I hope you are in this case wrong.
I think you already think you know the correct answers. The institution's only purpose will be to add legitimacy to what you think you already know, so you can point to it and have others listen. I highly doubt you think there is any chance of the institute finding that praying to thor heals the blind while praying to jesus heals lepers, or whatever other possible result you can think of.
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.)