lots of great advice you all have already said.
so there is not too much to add to this that is new, except, if you have a narrow bit of ground to get a mini or skid loader back to the tree, bring the loader in with an auger attachment, drill a deep hole near the dead tree, then bring in a utility pole (telephone pole) with the loader, plant it. friction saver attached to top of pole with climbing line before you plant it. Now you have a solid tie in, ..... but for only mid size tree heights.
another idea for non accessible backyards with a dead brittle tree and no other trees, buildings or high objects. if a field nearby, rent a hot air balloon pilot and balloon. haul balloon to work site with a skid loader as the anchor. Balloon floats over the yard. Climber repels from balloon basket and dismantles the tree.