I've very little experience craning shade trees. When I have, sling material seems adequate for smaller limb work.
When crane logging long evergreen sticks, the last few years the crane companies have alway had wire loop chokers, which we close with a shackle. They used to have the real heavy webbing, which we would undo from the ball, and install with no shackle required.
My log truck driver told me an incredible story. Some yahoo had somehow set a wire choker improperly around a small tree. The operator had not applied strong tension to set the bight. The cut was at the ground, the choker slipped off the end, the log fell and hit the crane cab, no one was injured. Phew!
Even more incredible was the crane operator's story. Another yahoo hotshot was ground cutting a large, long stick. Instead of making one cut straight through, he was fussing with some overlap mishmash. He kept telling the operator to take up tension. Anyhow, the 5/8 choker cable BROKE!!, leaving the tree teetering precariously between the house and the crane. Someone gingerly rode the ball up and quickly installed another choker and finished the cut. The crane operator was not without fault, I'd say, as he should have been watching his load scale.
I doubt if many of these "hotshots" are capable of determining the weight of a standing log very well. I'm pretty good, having a wood weight table , but a big stick did surprise me a bit recently. 23 inch top, 44 inch butt, 37 feet long, I thought 9000, it was 11200! No where near the crane limit at the extension, tho. Anyhow, two large firs, about 70-80 feet of wood each and a couple short hemlock logs, weighed 52000 lb.