Cap and the side of opposition are concerned with maintaining their privacy. So, okay, sure. Is there any real reason why superheroes could not privately register their details with the government in order to not be held accountable on a civilian level to the rest of the nation? Was there any reason for Peter Parker to publicly unmask himself? Clearly it served as a plot device to swell some drama between the opposing sides, but it doesn't seem like there was a real necessity to it. I don't see why we can't have our superheroes register privately in order to be held accountable on a government level for any misdoings or accidents or so on instead of risking an omission of privacy to the public.

I'm not familiar with the intimate details of SHRA, just trying to arrange a compromise here for fictional events which would work for all parties!