Obviously, people don't like invasive data collection. People don't like wars either. People generally have the kind of common sense that institutions and businesses lack.
My feeling is that you don't win fights like this by making a principled stance, then trying to get the public on board. You do it by finding wedge issues, where the argument is so strong that opposition is very difficult, then using victories there to build momentum for the next fight. That's what the civil rights movement did. That's what you have to do if you're fighting from a weaker position, and I think privacy will always be a weak value in western-style democracies because there are just so many compelling incentives for actors to erode privacy, and the threat of losing privacy is generally abstract, and only felt by already marginalized groups.