The explanation on the set trigger I used was for a modern set trigger. With those you can set the gun off by pulling the front trigger without setting the rear trigger. Some old original locks did not have a "fly" on the tumbler or a half cock position. The fly would cam the sear past the half cock notch when using a set trigger. By eliminating the half cock notch you didn't need the fly. The set trigger also has fewer parts on old originals.
Your rifles sound real nice.
I have two that work the way you described. You have to cock them before you can set the trigger. 30 cal sounds right for an original that hasn't been with. Mine are 32 and 38 cal. Small cal is where the term squirrel rifle came from.
You don't have to restore anything on the lock. It looks fine and is in good shape just lub it.