Any crimes that the politicians are involved with probably do carry significant jail times, given how corruption and pedophilia apparently still exist.
> 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
I don't really see this playing out in the politician's favor.
Politician: FBI is blackmailing me! Press: with what? Politician: ... Press: do you have any proof it's FBI? Politician: Nope FBI: We're not doing anything FBI: (anonymously releasing some info) Politician: resigns / get jailed Press: boo bad politician!
Or:
Politician: FBI is blackmailing me! Press: with what? Politician: They say I stole a billion dollars FBI: we're not doing anything, but we should investigate this FBI: wow this guy really did steal a billion dollars, we can't ignore this Press: politician announces he stole $1 billion dollars, gets jailed
I don't get one thing: if politicians' decisions can be purchased by campaign donations and lobbying, why would they be any more resistant to something more persuasive? Again, to reiterate, blackmail does not need to be letters with threats compiled from newspaper cutouts. There's a whole range of methods they could be using that rely on the information they collect.
I'm not taking a concrete position on the question, but given the secrets that plain ordinary people keep, I would not be surprised if the data available on people is used for coercion and nobody talks about it.
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.