While on one hand it's a bit of a side show to argue about this, it's also far more likely that people in general will see the problem if they see officials repeatedly being wrong or outright lying about abuses than to get them to accept there's an issue with "just" collecting the data.
The "just" collecting the data part of the problem is rarely seen as an issue with those who largely trust the government, as it is hard to get them to understand that even if the current government is ok, things can go rapidly downhill as they tend to write off any examples as too extreme or "can't happen here".
The best way of reigning this in, is unfortunately to focus on any too close for comfort abuses. E.g. if one found NSA staff stalking ex'es would likely get a far more furious response than NSA staff collecting data about large swathes of the population.