The phenomenon described is not due to the Oberth effect.
What would be happening would be that the human in the spacecraft, and the spacecraft, are accelerating at exactly the same rate because they are accelerating due to gravity. Thus, the human feels no pressure accelerating him (no outside force acting upon him) from e.g. his seat. And his internal organs feel no pressure accelerating them from each other. They (the craft and the human and all his internal organs) are in free fall together and feel no forces acting upon them despite the whole system (craft-human-organs) being accelerated to tremendous velocities.
What would be happening would be that the human in the spacecraft, and the spacecraft, are accelerating at exactly the same rate because they are accelerating due to gravity. Thus, the human feels no pressure accelerating him (no outside force acting upon him) from e.g. his seat. And his internal organs feel no pressure accelerating them from each other. They (the craft and the human and all his internal organs) are in free fall together and feel no forces acting upon them despite the whole system (craft-human-organs) being accelerated to tremendous velocities.