I have similar first hand experience with this. Big oak split and one whole spire went down on twenty by forty eight woodshed that was of little value. It was old and decrepit BUT I had replacement insurance on it.
Spire was separated at base of tree. body and top of tree were on the shed which housed an L 260 Kubota Tractor at the time along with mowers, tools, wood, tons of stuff that was valuable to me. I phoned the agents office on a Friday and left message. Didn't hear JackSchitt from em. I was fuming. I took photos of it all with my digital camera and began some of the removal of debris. It was clear to me from professional experience I needed a crane to get the tree remains off the building to save further damage to equipment, and not get hurt, myself.
I worked all weekend doing what I could. Monday I phoned the agents office and told them in so many words to get their butts in gear and come assess my problem.
The guy apologized up and down, knowing how mad I was. I presented him with photos which showed the removal work I did over the weekend and said I needed a crane. He balked. I said "fine, dude. You can figure on buying me a new tractor and a new mower amongst other things". His tune changed. Long story short, they sprung for an aborist crew, Tandem axle Pete with a crane to remove and clean up the debris. Tractor was saved. Only gear damage was complete hood assembly for LX series Deere riding mower and a metal gas can.
They cut me a check for about fifteen thousand to replace my shed. I was satisfied. So, I say, take your pictures in process. Document your time and go ahead on it.