How does not signing up protect your privacy? The only information the government can glean from your application is your SSN and the number of kids in your household. So what?
The primary point is that money not paid out in prebate checks ends up funding the government. A secondary point is, where you say, "so what", to giving up that amount of privacy you do not speak for everyone. Many people will chose not to divulge that information.
I find the amount of information about me in the hands of the IRS under the current system very invasive, but telling them my SSN (which they already know) and the number of people in my house, no big deal.