At some point between the period I mostly study (up to 1830 or so) and the First World War, military establishments started treating what we'd now call "staff work" as a valued professional discipline. At what point did staff colleges, studying battles and battlefields, and planning for future engagements become routinized, and who led that charge? In my head, I've always associated it with the Prussian military, but I have no particular actual knowledge to explain why I think that.
It is tied to the Prussians, as they had already established staff colleges at the time you study. Clausewitz was a graduate and future instructor. The practice began to spread following the success of the Prussians in the Franco-Prussian War of 1870-1871. Most major armies, including the United States, sent observers to the conflict and saw the superior way in which the Prussians conducted integrated their cavalry and artillery with infantry and their superior logistical operations. This led to widespread emulation of the Prussian way and furthered the widespread popularization of Prussian military works, including On War.
In 1878, Emory Upton, then Superintendent of West Point, wrote The Armies of Europe and Asia, following a tour of Europe as a report to General Sherman and argued that:
The perfection of the German military system lies less in the military organization than in the exactness with which men of every grade, in every branch of service are trained for the efficient performance of their duties.
He then goes on to detail the education of soldiers of every rank in the military, but highlights the general staff school, or War Academy, for praise, stating:
one of the best tests of the system of German military education lies in the fact that, at the outbreak of the Franco-Prussian War, every general in the Prussian army was a graduate of the War Academy.
In 1881, Sherman would establish the Army Command and General Staff College at least partially based on Upton's recommendations. Upton would spend the remainder of his career advocating for military reform in the US with various degrees of success. One area he most vocal in was calling for a better system of education for officers, arguing in the posthumously published Military Policies of the United States that
In every military system which has triumphed in modern war, the officers have been recognized as the brain of the army, and to prepare them for this trust, governments have spared no pains to give them special education and training.
Frustration with his own lack of advancement, he was a brevet general during the Civil War but never regained the rank after the war, and frustration with the failure to implement what he saw as essential reforms led him to commit suicide.
Edit: Fixed a couple biographical details of Upton I had wrong