How would one extract a brick chimney embedded in the middle of a house?
We removed an old brick chimney to install a modern (well, 1970's modern - stainless steel is more common these days in your better chimneys) block and tile chimney by simply hammering and removing bricks. If you do it top down it's "apparently more tedious" but actually less time overall than anything involving "knocking it down" and then clearing up the mess, when it's inside an occupied, normally clean house. A wide mason's chisel may help. Bricks are actually pretty easy to get apart when you learn how, and that won't take long if you pay attention to what works on the first few layers.

Incidentally, if you drop the bricks down the chimney they pile up in short order and become a relative pain to remove. Take it from a former teenager who thought that was a great way to get rid of them until we hit the pile coming up as we were tearing down.