No, surveillance is not a threat to democracy, people not caring about democracy is what threatens democracy. Democracy has never been a right, it's always been a privilege that people demanded, and guess what, when people stop demanding it, power will naturally collude.
I agree with your point that apathy is a threat to democracy, but there are multiple threats to an open and fair system, and a surveillance state is one of them.
If the surveillance state existed before the democracy and continued on without the consent of the governed, then it wasn't a democracy to begin with. If the citizens allowed a surveillance state to take root or their vigilance slipped, then continued vigilance would keep the surveillance state in check. However, I'm making the assumption here that for the surveillance state to take enough power to make it unstoppable (without violent revolution), it would have to grow to the point where its secrecy is no longer maintainable, and the aforementioned vigilance would take over and bring in the rains.
Now that I've written that, it all seems like semantics...