The big ones like Mayflower, Allied etc carry the best insurance and actually train their people.
Yep, that has been my experience also. In order to get full insurance coverage though, they need to do 100% of the packing, which may not cost as much as you might imagine.
One thing way that they can do the one-room move is bring large wood crates and pack the stuff in the crate. This way, there is not a way for this small load to get tangled up with a mixed load of other stuff on the truck.
I used this method once myself for a 4 bedroom house of stuff when there was a gap of a couple of months between when I needed to move from the old house and when the new once was available for moving into. The crates were put into warehouse storage until the moving destination was ready to go.