And for example with the Spartans I don't think there was too much formal education about the "finer points" of practical campaigning; not only was Greek warfare for the most part somewhat too straighforward and short-duration for that to be particularly important anyway, but such things in general were generally learned by practice and maybe watching senior commanders do stuff.

The whole idea of formally and systematically training an officer corps to wage war "scientifically" is in general quite modern. On the whole I don't think it really turns up before the "pike and shot" period and even then rather late, more towards the beginning of the "musket and bayonet" era. The Prussians tend to get referred to as the first ones to systematically educate their staff officers, AFAIK.