As my interest is primarily in the Army of the time period, I'll reserve my comments there.

Yes, the Engineers and Artillery were the only ones who had any real "schooling" in their "trade" (as the other two branches viewed it). By and large, a great deal of the training of Infantry and Cavalry Officers was, as you said, "picked up". Bits of regulations and drill can be found here and there, but there wasn't any single book that laid it all out for them.

(This has made re-enacting quite difficult at times. We have the basic drill manual, but beyond it is part detective work, part guessing game. Most often a reference will say something is to be done "in the usual way." Every few years some new primary source is uncovered and we have to retrain everyone in some aspect.)

As mentioned, officers were meant to be young men of good families who had a stake in the nation's interest (i.e. property); and as Wellington did note, "there's nothing so stupid as a gallant officer". Nor is a mess likely to be the most conducive environment for intellectual stimulation. There were undoubtably many for whom anything beyond food, drink and games were a waste.

On the other hand, many other officers were quite well read and would share their library amongst their peers. There are references to amateur theatrics put on where the army was posted for some time.

One might argue, that this might represent a very active minority. Both extremes, and all in between, can be found in contemporary accounts. I could, however, rather cynically question how much of that "intellectual" conversation could be found in the boardroom of a modern corporation. ;-)

I remain, etc.
Jason E.