So the decision was made to remove the tree. We started by removing all the lower limbs, which just confirmed our belief that the tree was dead.
Then we took the top off.
This produced a large stack of wood that we gave away on Craig's List, and an even larger stack of small branches and twigs which the City took away in Large Brush Pick-up.
The remaining part of the tree -- the trunk -- was 12 feet tall and 60 inches around (at the base). It took some 3 hours of work to remove the top branches and cut down the tree.
This leaves a stump
but the yard is now clear of the tree and we can start digging, to remove the stump and any rock under the dirt.