You can not just treat the pitching moment independently...
Extending flaps also results in a higher lift coeficcient of the wing, so either speed will reduce, or AoA will decrease. AoA decrease in combination with reduced L/D (reduced glide path) will both bring pitch down. Especially on aircraft without leading edge devices, this nose down attitude may be quite impressive.
You also have to see the full picture of pitch being the sum of the flight path angle and the AoA, so a nose comming up or going down depends on both. The flight path angle is affected by engine thrust, just like the pitching moment.
With all these aspects, it becomes clear why every aircraft has individual ways of reacting to flaps.
Typically first a tiny nose up followed by nose down.
Sounds like pitch up due to the downwash on the stabilizer being a little stronger than the pitching moment, followed by a reduced glide path angle due to higher drag...