I think that "post" to the left of and above the switch on your last photo (there's a red arrow pointing to it in my first photo) is supposed to move toward the long switch leaf when the plunger pulls in, and open the switch. It's labeled "lockin switch" on your score reel, so it's in the locking circuit of the score reel and when that switch opens, it cuts power to the coil.

Your long blade looks a little mangled, so you might have to straighten it to get it to work correctly.

I have a 1965 Flipper Pool that has the same configuration. When I push the plunger in with my finger, that post opens the switch (lower photo).

FP score reel 1 (resized).jpgFP score reel 2 (resized).jpg