My watchmaking school days are far away but I reckon that it's not a "second hand" that you have in the center but the chronograph's hand.

When you press the button, (1) the start/pause cam turns one increment, (2) the chrono's cam, pushed by its spring rotate in place (3), and put its wheel ,which is connected to the main gear train and is always turning, in driving range of the central hand (4), causing it to turn (5).
Pushing the button again will turn the start/pause cam an other increment, pushing away the chrono's cam, and thus the central hand will receive no more input force.

You can see a gap below the chrono's cam. It's there to allow the axle to move.

You can't press the button because the start/pause cam can't overcome the tension in the copper wire. My theory is they blocked it because the chrono would often trigger in the pocket which puts a higher load on the spring, sometimes enough to stop the balance from oscillating and stopping the watch.

I don't think anything is broken with it, but like other anons says, pocket watches with a chiming movement are worth quite a bit of money. You have one beautiful watch on your hands.