Automatic trim adjustment does not push down the nose aggressively
Given a plain canvas at Boeing and the desire to extend the life of the 737 what they have actually done is upgrossed the capacity and length of the airframe and added additional power for the 737 MAX. Since Boeing was using the basic design of the 737 they probably performed the modifications using a series of supplemental type certificates (STDs). This reduces the time and cost for the development of the new 737s.
Fundamentally they increased the capacity of the aircraft to carry weight, fly safely at speed, and be controllable across all flight surfaces and axes.
In changing the length of the aircraft AND increasing the wing size AND AND...changing the engine type, critically the center of gravity was moved (I am guessing) aft. While load and balance can control nose up (passengers and luggage-freight) making it safe to fly, the pitch axis (a sudden pitch downward in climb out is what happened in both crashes) seems to be what the software system was designed to compensate for in the MAX versions of the 737.
The reason the plane can exist in its upgrossed size however is the LEAP engine, for both commercial and aeronautical reasons. There must be something bad inherent in the design of this upgrossed version of the aircraft which the software control is meant to mitigate.
The LEAP engine is being operated commercially at less than maximum output in order to allow for a longer engine life, and the turbofan has some considerable novelty built into it which increases the thrust to weight ratio. Since there are conditions when thrust decouples from airspeed, for example clear air turbulence, updrafts-downdrafts, and since the engine seems to rely upon available air (quantity-density) for full blade inflation (operating at its rated capacities) and predictable thrust, maybe the software was developed to compensate for blade deflation (and the resultant reduced airflow across the wings caused by reduced thrust plus the blade re-inflation latency). If the air supply faltered in climb out, and the blades deflated, you would want the nose to drop quickly to reduce the angle of attack to prevent a stall of what are probably super critical wings, close to the ground. No recovery from that.
I wouldn't be surprised if the weather was really hot upon departure for those (now) two downed aircraft...the air is much less dense and stable and therefore more difficult for the engines to 'bite'.
The fact that the software interfaced with a single sensor in the two downed aircraft demonstrates a breakdown in communications between software engineers and the aeronautic wonks. Properly developed software, given the critical nature of its purpose, should have been relying upon multiple points for data for it to fly safely if the design flaws above exist as stated.
I think the point of failure was in the design of the aircraft-engine pairing given the 737 MAX is just an upgraded version of an existing aircraft, and that it operates at the edge of the performance envelope in its upgrossed design. The software was meant to make the design workable.
Properly operating software will make the plane safe to fly in automatic mode, however the plane is still flyable by a pilot (I am relying on Richard S. Bach and his Jonathan Livingston Seagull to explain what pilots can do to manipulate safe flight that machines (aircraft) can't do alone) manually.
Finally, and commercially speaking, the 737 series is a very practical solution for a range of flight requirements. Because of the similarity in flight characteristics between the different models and configurations, a pilot checked on type of one model can pick up and check out quickly on the other models as well. This is a big selling feature of the 737. It may be that the MAX series requires special training and increased awareness of the people that schedule and fly them.
Calling the nose down feature of the software control system a 'trim adjustment' is meant to minimize the importance of the role Boeing plays in these two crashes, I think. The software is designed for disaster prevention, but poorly.