I think this was intended to be a Presidential power. Invasions and Rebellions happen quickly. To expect a voting body, which may not be in session, to act is ridiculous. Only an executive body can act quickly enough fr these two causes.
“Clause 2. The Privilege of the Writ of Habeas Corpus shall not be suspended, unless when in Cases of Rebellion or Invasion the public Safety may require it.”
Oh, please. Lincoln was supposed, in an emergency, to call Congress back into session. He did the opposite. He waited until the spring session ended, at the end of March 1861, and then he made his moves on Jeff Davis by sending men to reinforce Sumter while loudly declaiming that he was doing no such thing, that men would only be put into the fort if this and that happened (yeah, right, Abe!), and so on. He did not call the Congress back for months and months, while the opening battles of the Civil War were fought and Lincoln transacted coups d'etat against Maryland and Missouri.
He did it again in 1865, when the war was ending. He waited until Congress was out of town to unveil "presidential Reconstruction" because he wanted Congress to have no part in it -- in peacetime. (So much for "these were emergency war measures!" and all that.)
It was a consistent pattern with Lincoln. His first inclination was to act through and by his own office only, and leave the Congress and the People trailing in the dust.
"The Public Safety" did not require the suppression of civil rights and habeas corpus in 1861; there were plenty of ways to deliver troops from New York and Boston to D.C. -- the Anacostia Naval Yard was near the Capitol and in full operation. The "emergency" in Maryland was fake, and the Marylanders knew it; it was a political confrontation and excuse to act tyrannically using the Army to suppress a State. It's as if Obama were suddenly to roll U.S. Army tanks onto the grounds of the State Capitol in Austin and send Army officers to take over the State government, Obama having unilaterally having proclaimed Texas to be in rebellion.
Think about that one for a while.