Have the Democrats ever put any power-grab instruments back in any boxes? I don't remember a time they have, and I don't think they'll start now. They are meek cowards.
They joined Republicans in limiting presidential power after Watergate. Granted, these limitations usually come after gross abuses. But these are gross abuses, and there's no reason to think they won't get grosser.
New York neutered the power of appointed officials and authorities after Robert Moses.
As the other poster mentioned, the post-Vietnam Church committee was a democratic party-led, bipartisan committee that effectively heeled the CIA and exposed many of the pretty dark abuses committed in that era.