I'm not covering any new ground here.
Proper gun safety by all members of the hunting party is vital.
Even if verified unloaded, always point the barrel in a safe direction.
As for the bolt up method of a "safety"...
I'm not a fan because there is just too much to go wrong.
It gets caught on brush, the shooter falls...etc.
Ultimately it will fall to your PH as to what he would like.
Our PH was comfortable with us and our firearm safety.
We loaded the chamber when we got off the truck and put the safety on.
Unloaded getting back on the truck or when the animal was down.
As
@Red Leg stated, this is where the Blaser R8 and similar systems make a heap of sense.
Having the chamber loaded and rifle decocked is about as safe as it gets with one in the pipe.
Being able to silently cock the rifle just before the shot is the best part.