That is an early Syracuse L.C. Smith from around 1884.

Yes, that is the way they did it.
I tried the Hunter Arms way with the punch and screw driver once. I didn't like it. If you slip the screw driver can slide across the metal and gouge the wood finger. The workers did this day in and day out and it was easy for them.

If you rotate the pin on the joint check to 5:30 and align it with the slot in the receiver it should go in the first time.

I do like how far these guns open when you break them, makes shell extraction very easy.

David