If Congress demands it? Yes, they can brief Congress on what they're doing. They were created by Congress and the career positions continue to exist at the pleasure of the Congress.
Further, there are procedures that exist to do so. Sealed sessions have been and are used for sensitive subjects to avoid inadvertent disclosures.
"All 535 publicity-seeking chatterboxes" are what we have. Their existence, their ability to speak freely without fear of prosecution, their diversity of opinions & constituencies, and their power to change things, function as a crucial check on the power of the executive branch. As representatives of the people it IS THEIR JOB to ensure that large secret programs that the public would never approve of, or which violate their oath to abide by the Constitution, are either reformed adequately or leaked, neutralized in some way, cancelled, or never funded in the first place. If they fail to do that for long enough because they've decided they're not able to, or because they don't want to know, then this all falls apart and becomes tyrannical the first time we get a President that wants power.
Notably, it ceases to be an option for Congress if the objectionable business is objectionable because it's a spy program that been turning Congressmen into blackmailed sycophants. That is hardly out of the scope of the technical power we have granted the NSA.
If they have created these programs and they've started to take on a life of their own, to the point that Congressional oversight and regulation is not adequate to control their activities (and yes, it appears that is what happened), then we have one option: Kill the tumor with radiation and a clean scalpel before it metastasizes further. Defund, publicize, interrogate, leak the objectionable programs, and end them. Maybe catch some loyal civil servants who were just following orders in the crossfire; But that's how it goes - they became an enemy, foreign or domestic, of the people and their Constitution. Too bad.
Sure, we might lose some tissue, but it's a damn sight better than letting a malignant cancer of our democratic system run wild.