Take your favorite symetrical wing aerobatic airplane and make it more nose heavy than normal, now adjust the elevator trim to compensate for that nose heaviness during normal (1/2 throttle) cruise. Now see what happens when you add throttle. That's right, it climbs! Now shut the throttle off and, voila!, it goes into a shallow dive.
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Um, yeah, but that's like cutting off your feet so you don't bump your head in a short doorway. It would not be an efficient way of inducing lift. Movable flaps would do it without inducing drag at the higher speeds a symmetrical is designed for.
A coincidence does not always mean a cause and effect. We usually only find flat bottom airfoils on trainers and symetrical airfoils on aerobatic airplanes so we assume that the airfoils give these planes their handling properties.
So you're convinced it's a coincedence trainers have flat bottomed airfoils? A flat-bottomed non-tapering airfoil is also almost impossible to stall into a snap roll, they just drop at the nose. Yeah, I could also give my horizontal stabilizer several degrees of negative incedence and my symmetrcal winged planed would be throttle sensitive to lift. But the drag would be fierce and it never would fly well level.
Symmetrical wings give wide speed envelopes without needing trim adjustments. Trainers have no need for wide speed envelopes, so the advantages of the flat bottomed wing make it a good choice.