Where does one push on the world if one wants to make a large positive impact? This entire mission is dangerously dubious, but we’ll assume it’s a good idea for now. The “positive” part is hard, and a matter of case-specific judgement, but we can briefly comment on “large”:
Generally, to make a large impact, one wants to work on things that are “upstream”. That is, things that influence and affect other things. Things that have their effects multiplied. The “elites” are upstream of a lot of things in society. These are the people who build things, run things, maneuver into positions of power, lead, and determine the direction of a society.
Because of their position, if they are sane, well educated, fertile, cohesive, and organized, as in the early british empire, the proles and other assets of society will be well-managed and well-utilized, and you will get empires, science, great art, law, and general flourishing. On the other hand if the Elites are decadent, disorganized, barren, or insane, as in the late roman empire, and disturbingly in our own empire, you tend to get bad results; they have large capacity for destructive behavior. Things have not fully hit us yet, but the signs are there that our elite is crazy, and the precedent is disturbing.
So if one wishes to build and maintain empires, it is probably necessary to build and maintain a competent elite class. The rest will manage itself (or rather, will be managed by the elites). If one wishes for certain ideas to be utilized in the administration of society, it is probably necessary to teach those ideas to a powerful elite. If one is able or needing to route around the “elites”, this is a bad sign.
It is true that a competent elite will not be able to accomplish much without a strong economic base, geopolitical opportunity, a virtuous and capable prole class, a sane worldview, and a good political system, but one does not simply conjure such things out of thin air without a competent elite. Only an elite class is in any position to construct such things, and even they will not always find it possible.
Therefore the primary mandate for a strategically-self-aware empire is to maintain good behavior in the elite class. If it is necessary to manage the behaviour of the proles, it is ten times as necessary to manage the virtue and competence of the elites. It is also easy to get wrong.
Therefore a major strategic focus for any ambitious school of thought is to engage with, influence, and assist the elite class.