Should gun rights advocates promote relating privacy rights to gun rights, I suppose that could spoil the perception of privacy rights among large numbers of US voters.

Maybe, in that scenario, gun-rights voters end up carrying privacy-rights US policy influence, but then we might also have a lot of other US people thinking privacy is not for them, in their own day-to-day lives.

Politics and perceptions are way outside my expertise, but things like this sometimes come to mind when working on technical stuff with privacy&security implications.