] The only Army officers who received any kind of intellectual training in this era were the engineers and gunners who went to Woolwich.

Not quite. The Duke of York established military academies at High Wycombe in 1799 and at Great Marlowe in 1802. By 1815 these establishments had turned out 114 staff officers and 651 line officers. This is somewhat less than 10% of the army establishment, but it is something.

YHOS,
Snarkhunter