It is worthy of note that some (myself included) view the National Security Strategy, the highest-level strategy from which other strategies in the Executive Branch are derived, as a document stating aspirations and a tool to communicate indirectly with other nations rather than an alignment of ends-ways-means in the military strategy sense.

I would posit that the Spanish-American War, giving us PR, Cuba, and the Philippines was our real first look outside our borders in a deliberate and semi-strategic way. It was WWI that led us to realize our need for an external strategy since it not only established the U.S. as a power player in the world (as you said) but showed the impact we could achieve by mobilizing national power in a cohesive way.

But who is this in America? It cannot be the President alone, though they are the ring leader of the group. The NSC, maybe? It would give buy in to the strategy if that group developed it with some level of consensus. Whoever it is, they need to be accountable for it but also limited in exposure to political whim.