The MACS does not activate by pointing the nose down, at least not directly. Upon receiving a high angle of attack signal from an Angle of Attack sensor, it trims the stabilizer downwards. It does not override the pilot's imput of the elevator. But in these 2 cases, there winds up being so much downward stabilizer trim, that there is unsufficient elevator control to prevent an uncontrolled pitch down.
Pilots are trained in all kinds of systems failures. Boeing didn't think it was necessary to train them in case of this occurrence: that of the AoA sensor being faulty and sending a false stall indication to the flight control computer.
This problem was not insurmountable. The day before the LionAir crash, another crew flying the same airplane and experienced the same problem. However, they had the presence of mind to cut the power to the stabilizer trim, as is the procedure for a runaway stabilizer. They weren't specifically trained for that; with proper training, I believe that both accident crews would have reacted appropriately.
One has to be careful when using such layman terms as 'pilot error'. In reality, the pilots screwing up is the final sequence of events that result from poor aircraft design, poor procedures and poor or inadequate initial and recurrent training. Flying a transport category turbojet is not like driving a fucking car.