I can imagine how a registration act would work in the DCU.
If you look at the original Suicide Squad title, you just know the govt, or certain people in the govt, would use secret identities as bargaining chips & blackmail tools.
Remember how Waller said she had Batmans fingerprints and would reveal his identity if he didnt leave the Squad alone!
There has always been corrupt members of the govt & military within the DCU, and these people were always trusted with top line jobs & info.
Wade Eiling is another name that springs to mind!
A registration act is not about whether the heroes trust the govt, love their country or worship the pres, its about individuals in the upper echelons of govt, and can they all be trusted!