You don't need to remove the rivets - and (sorry Ron) don't listen to Ron - I have never known an SF starter freewheel plate to come apart and I have seen and worked on MANY over the last 45 years.
You may find that the piston/s is/are stuck inside the pesky hole. If it's really bad, it is possible (I have done this recently) to drill form the outside into the cavity and use a prod to remove the dodgy piston an d it's broken pieces of spring that are no doubt blocking its movement. It isn't easy to get the angle right and I used my lathe to align it, using a mill vice on the cross slide, and milled a flat at the drilling point (the curve won't allow a drill bit to bite without the flat starting point). I did it for all 6 pistons. Worked a treat and there for any future probs.
As you've discovered, it's difficult to get them out from the open end. You can try tapping the freewheel against a piece of wood in an attempt to let inertia force the piston out, but be careful not to whack it too hard as it can crack the insert (a replaceable part, but preferable to avoid).