Devils advocate here, but I don't think government entities can do free/unpaid appearances. I think it crosses the line of looking like a state takeover, so it's a mandatory that they pay market price for rendering services of any kind. So yes, while they paid the Falcons 850k, that is technically the private market's fair market value of hosting a ceremony of that size/length.
That doesn't mean the NFL isn't taking advantage of the Iraq War, Afghanistan War, and Islamophobia, but it should be pointed out in case people start complaining about the government's role in this.