It's not an old chair and by the looks of the staples someone has done this once before. Who knows?
If you are redoing the seat I wouldn't be afraid of destroying that particle board seat frame.
If you absolutely can't figure it out after removing the seat frame (bottom, whatever you want to call it) then make it so it folds in like a hinge and attach stiffeners to hold it flat once wedged back in place.
The older method of supporting the cushion with webbing would make it a lot more comfortable.
Note, I don't do anything with upholstered furniture.
You could eliminate the springs in the above diagram. In fact, 3" is as short as they come and you'd really have to come up with a scheme to fit those.