I had this disappearing oil problem with my first Toyota (1974). New oil is Clean, and therefore very hard to see..even on the dipstick. Leaking oil on or under the engine will be hard to SEE.
My '74 Toyota had a mechanical Fuel pump on the side of the engine which is driven by a lever which sticks inside the engine and gets pushed up and down..ie pumps Fuel, by the cam shaft. This engine was NOT an overhead cam engine..but rather..overhead valve.
The short story: A gasket in the (mechanical) Fuel pump had ruptured and ..as well as pumping fuel on one side of the diaphragm, was pumping OIL out the breather hole on the top of the fuel pump. The clean oil was blowing in the wind back under the car and was pretty difficult to see. When the car was in the garage..engine not running...no leak.
The Toyota dealer wanted to replace the oil pan gasket. He didn't want to look for anything else AND wouldn't guarantee that this would fix the problem because he was guessing.
I suspected the fuel pump and he laughed at me. I went home and removed the fuel pump and took it apart. Voila. I took the pump back to the dealer to show him the pump internals..and to buy a new pump. I thought the mechanic would be interested in LEARNING something. NOT so.
The New
Fuel Pump fixed the disappearing oil problem.
NB