I just looked at several Blinker lever switches that are removed from the column. The hole IS open all the way thru.
I had a couple of these and they were made differently. a couple had the back-side of the hole plainly visible - but there was one style that has the hole kind of "blocked" by a piece of plastic. A striaght punch would not work - but it looks like there is enough space betweeen the plastic and the back of the hole to get something in there to push out the broken piece of the signal stalk.
You might need to remove the steering collar to gain access to the back of the switch - but the back of the hole IS open !