Often people confuse the "design manoeuvring speed Va" which is a FAR/CS-25 structure certification definition with "flap manoeuvring speed" which is a OEM definition of an airspeed which allows for "safe" manoeuring (usually 25º of bank with a 15º overshoot allowed) without stalling. The former is a maximum speed whereas the latter is a minimum.
In terms of the design manoeuvring speed, Va, I don't believe OEM's of large jets usually informs the pilots of their value because they don't really need to know. Same as Vc, Vd and other structural airspeeds. We stick to our operational values, VMO, VNE and so on, these all have healthy margins baked in versus the structural airspeeds for all flight conditions.
Below Va, an airplane will withstand a single full application of a control surface, but not necessarily a repeated, oscillating application which is a completely different structural problem. Each cycle of control application increases the loads on the fin because the sideslip angle becomes larger and larger with each swing. While the training the F/O of AA587 underwent maybe didn't highlight the dangers of multiple, repeated and alternating control input applications, I also don't believe instructors were actively training students to react this way. It feels like the overreacting tendency of this F/O fell through the cracks without anyone noticing or bringing it to his attention to correct it.