My point is that those charastics aren't due to the wing having a flat bottom. All airfoils increase in lift as airspeed increases, even symetrical ones.

What is more often the case with trainers is that the wing is set at a rather high incidence, plus a fairly far forward CG results in the pulling out of a dive you describe. You can cause the same effects with a symetical airfoil, and you can make a plane with a cambered airfoil (such as a "flat bottom" airfoil, which is just another airfoil with camber) NOT do those things.