Re: "the aircraft’s autopilot system will trim against the control column force"
On many light aircraft the autopilot doesn't control the actual control linkages (Which are mechanical) but only the trim system, which has enough control authority to perform normal "autopilotable" maneuvers.
In this case the pilot pulled on the yoke causing the nose to rise. The autopilot doesn't know better than to try to correct this so trims down. The pilot probably instinctively just pulls a little bit more on the yoke, but this cause the autopilot to trim down even more. The elevator loses a lot "nose up" control authority when the trim is full down. So it might nog have enough to keep the aircraft climbing at lower speeds. When the pilot relaxes force on the yoke the aircraft immediately goes nose down into the terrain.
This could have all been avoided with proper training for use of an autopilot on a light aircraft, and the TO is as much to blame as the pilot.
