No, temp duty at the rank is still not the same as going to a school to learn what is needed at the rank. Also the Staff schools are where your new doctrinal ideas typically generate.
First how about some definitions/qualifications . . .
Lieutenant or entry level officers, learn through their academies how to operate their small unit- platoon/lance size. They are then trained at their posting by the senior NCOs and the CO/XO when they have time- who usually tell them to listen to the experienced NCOs since that butterbar is clueless. The LT spends his/her time learning how to command the platoon- hopefully not micromanaging (good luck on that, its a phase and some never get out of it- they make good staff officers). They will then either be rotated to another platoon (usually some form of support/weapons) or sent to be a go-fer for a battalion or larger (BDE/RGT) staff. The idea here is that while they fetch water and spend time worrying about updating maps they also start learning the basics of how larger formations operate & the complications of those larger formations. Then a stint as a executive officer will roll around, where you study under a (hopefully) excellent company commander while managing the unit along with the company's senior (or First/Top) sergeant. Provided they show some ability and things progress, they are then advanced to commanding a company- where knowing some of what goes on at higher helps make the company perform. The company is a lot like the platoon only larger . . . but the difference between company and battalion is a big jump.
With command at the battalion level comes a whole lot more discretion about logistics, transport, commo, and all the other details the BN sets for the company . . . that the company commander never had to set up. For instance, the company commander for a route march from a assembly area to a jump off point in the AO just has to ensure the movement briefing, order of march, make sure commo is up, and see that the company moves off on time.
Battalion staff has to organize-
order of march between the components
time table to stagger component's movement (A Co leaves at 0915, B Co sets out at 0940, etc)
plan any refueling stops if required
plan for any rest periods/driver swaps
coordinate movement with higher
hand off control if replacing unit
divided assigned AO between components
get latest intel update and push it to components
perform any liaison work (ROK liaisons for instance when operating in S Korea)
None of which a company command really prepares you for . . . and being a captain on a BN staff does not prepare it for you either, even if you hold one of the big slots like Ops or Intel. To get a idea-
http://www.cgscfoundation.org/about/the-college/Additionally, I THINK the US Army also has big regular exercises carried out by simulators at Ft Leavenworth that use the officers attending the school but also bring in troops at all levels to participate. One of my buddies volunteered for TDY attending a exercise at Leavenworth- I want to say it was a big theater war exercise. But he manned a panel to be the FDC input.