The OP's video was a clear case of F ups all the way round. You can guess even without looking that the tree was rigged with chain or rope and choked on the back cut side, so all the lift was pulling it off to the side. It wasn't too big of a take for the crane, but when you're going to do big picks like that you need to balance your rigging carefully, and have multiple rig points. 2 or 3 tie ins would have been fine. There are some self balancing ways of rigging these also, having a rope through a porty rigged off the crane as the second point so a groundy can adjust it for example. I dont think the crane op had too much pretension, it as just the angle of the rigging. That is a sign of lack of experience on behalf of the crane op though.
I've done lifts up to 80T Heavy equipment, not trees) with a couple 220T cranes, at the limit of lift on both cranes because the lift was so far from the cranes. You need to do lift studies and carefully balance your rigging on large or heavy lifts.
I prefer working with crane operators with experience, and prefer older guys because they have patience. Crane tree work can be slow, and young guys get impatient and jump the gun sometimes. When dealing with an op who is a bit trigger happy on the cable I use them as a static point and have them leave the jib of to the side. I rig, then fall the tree onto the line with minimal slack so as not to shock load. Then they can take up. No need for preload that way. The same rules for rigging apply no matter whether you're using a crane, a GRCS , a porty or a handheld rope.
Shaun