It's difficult for the FBI to use mass surveillance techniques to blackmail anybody. Sure they can make allegations, even true ones, but if they present evidence it would be pretty clear where it came from. Only so many entities have the ability to gather the kind of data necessary to separate fact from rumor.

In terms of incentives, blackmail is generally a lower tier crime in terms of likelihood of success. Think about if someone tried to blackmail you. There's a good chance you'd think "Sure I can give in to their demands now, but they'll just ask for something else later and they never have to stop asking". Unless the crime being covered up involves significant jail time, most people would rather face the consequences than live the rest of their lives under someone else's control. Many former politicians have successful careers even after mundane scandals (Weiner and Spitzer come to mind).

And depending on the nature of the secret, most politicians would get some slack from the public for having the courage to standup to FBI blackmail. They might even be able to spin themselves as a flawed hero. In fact, having a three letter agency try to blackmail you right now is kind of a get out of jail free card.

As tptacek said, this is really fascinating fiction but it's just wildly impractical in real life. Blackmail is messy and has a lot of moving parts that are difficult to control. The FBI is better off doing exactly what they do now: asking Congress for money and getting it.