This video is from a guy I know and many have ask him (on another site) 'how he does it' so he made a video.
If you have seen or read about 'power bedding' methods (years ago) or using Peat for trap bedding, he combined both for his style. I think he's using a 30p nail.
But making the trap bed slightly larger than the trap has always been the key to solid trap bedding.
The problem with bedding traps solid is usually related to digging too large a bed.
My traps are center swiveled. By that, the chain is on the bottom frame of trap, in the center. So the trap, when bedded, could rest on the chain and wobble. I solved that by digging my trap beds in a cone shape (V), where my levers and jaws are stable but a void where my chain lays. After staking, I may use hammer to pound out depression for my chain to lay in. At times, I may hammer the outside dirt of my bedded trap to make it a bit more solid.