Drywall – What’s the point of a thin (1/4″ width) piece of wood set diagonally between two studs

Do you have any high powered magnets? like the rare earth neodynium magnets? If so you can probably rub one around the wall until it attaches to one of the nails in the studs. Or you can try thumping on the wall, a stud will sound alot less hollow versus empty drywall. That's how I've searched around in the past.

It is not safe to simply cut a hole in the studs, and while the safe probably would transfer the load, you do not know that the safe was designed to transfer vertical loads, or how it will react. The consequences of a mistake here could be severe long term.

Therefore the proper course of action is to reframe the wall. Generally you would do this by adding a lintel (a RIGID horizontal beam that spans an opening typically over a window or door) at the top and bottom of the wall wide enough to transfer load to either side of the existing column, and then add new columns on either side of the existing post to carry that now transferred load down to the floor lintel and thus to the underlying support.