The 12 rail bolts go through the cross-member sub-frame, through the slate, and bolt into the rails. So the rail bolts are in fact the slate bolts. Take the rails off, and the slate is free to move.
Then who ever has been recovering that table has been using spray glue and gluing under the slate. Yep, you're going to have to try and lift one end straight up, if the table lifts up with the slate, try the other end. If still no luck, run a putty knife between the slate and plywood sub-frame and break the glue hold that way.