We all know and understand the ancient proverb:- "Necessity is the mother of invention". In other words, "innovation" (as in:- real progress) often happens when there is an imperative demand.
History provides many examples of innovation being driven by competition. That is, a struggle for advantage between competing organisations (countries, companies, teams or even individuals).
Massive centrally planned organisations generally lack much fertile ground to allow initiative to flourish. In fact they often stifle such efforts.
"Can you teach Innovation?" ... can you teach a mindset? I would say it's a bit like other personal attributes (such as "leadership") ... yes, you can "bring people along" (aka "mentor"), but generally speaking the most useful folk (that is, those most likely to make use of such qualities - to "make a difference") already have it. It's probably simply a matter of identifying people with such traits, and then encouraging them.
You mention "the system in which the target sits". Indeed; you've lost me there. Which system? What target?
