The assumption behind this effort seems to be a sort of retribution-based. So there's a couple things wrong with that - staff are the only ones using computers, not Members, and it doesn't bother us too much. We expect none of our browsing history to be private and are used to having every part of our jobs as public information - see Legistorm.
Members are not going to care if their staff's web browsing information is public - even if it did go beyond just who is visiting govtrack and when. They'd fire the staff they needed to without hesitation and still show no remorse for their vote.
This is without even mentioning that only Republican offices voted in support of the rollback of the FCC regulation.
It's a harder issue to solve, but the problem really is that business interests will almost always trump constituent concerns. Having privacy groups come meet with Congressmembers helps - setting up an organized campaign helps somewhat too. But for Republicans, this bill was only a matter of what made economic sense.