For person who is not afraid of Heights, a bucket truck might be do it yourself solution.
Inside these chimneys might be clay liner tiles.
I watched a DYI chimney demo on a 1930's one story, in San Antonio, with a fireplace below.
The fireplace it protruded from the outside wall versus being an inside fireplace and chimney.
It was the owner of the house and a couple of neighborhood teens who handled the demo and clean up.
The owner went on the roof using a ladder and took a couple of wacks at the chimney with a sledgehammer and big chunks of brick, morter and clay flue fell down on the roof and damaged the roofing, and rolled off into the yard where the boys broke it up.
The put in wheelbarrow and took it to a pickup truck which squatted heavily when filled with bricks.
It looked like a dangerous footing situation to be on the roof but the chimney came down to about chest height that way.
Then he took the sledge and just bumped the bricks loose, down inside the fireplace.
He took the whole fireplace out so there was a hole in the wall (and added a window) and the roof had to be extended over where the chimney went through, and they had to replace the shingles that were super damaged from the initial bricks falling.
Permits, I don't know if that required a permit or not in San Antonio. In your area might though.
It was not a fast job and they overloaded the back of the pickup quite a bit weight wise. It was squatting down heavily but they finished it in one long Saturday with a lot of iced tea.
I have also personally seen two owners die from falling off of one story roofs- so if you do go up there be careful.
Good Luck!