I am thinking there has got to me more in there than dead birds. If you can't punch through it with a pole, I suspect there is something else, perhaps a piece of old cap, or a collapsed liner. I would keep banging the crap out of it until it broke through, then try to clear it out with a chain. If that doesn't do it, sounds like a pro might have to be budgeted in.
I think you are right. The first owners/builders were pretty meticulous about keeping things in order and they would have kept a cap on it. My dad and I have talked about maybe a cap falling down the chimney and dead birds on top of it.
But we are already pulling dead birds from the bottom - no more soot. A cap could be blocking portions of the thing.