Thanks for the video, your "be overly cautious" approach is the correct frame of mind for such potentially dangerous endeavors.
I was a trained and certified crane rigger long ago and wanted to throw in words of caution- when soft shackles get frayed the factory rating becomes meaningless. Ditto the web straps/ropes. It's really hard to discard such pricey stuff but it's better than the photos of the worn piece showing up in court during the lawsuit. Their ratings apply to a dead straight pull on the strap with both ends ideally attached. Any departure from ideal begins to derate the advertised capacity. Using a choke attachment, pulling at an angle, etc. all derate the factory specs.
I guess if all of the rigging involved is massively overrated for the job you can think less about it, but I'd like to know if anyone from the towing/rescue world knows the kinds of loads are involved (like from a winch with a load cell) in pulling out a heavy mud bound vehicle, uphill, and at an off angle? Not my area but I bet it can be many, many times the weight of the vehicle at some single instant during the job.
Be safe.