Forgive me, but when I first read your post it I thought you were talking about doctrines instead of division templates, which I thought was a great idea. I know it was not your intent, but it gave me the idea below.
Instead of focusing on division templates as a need for experience, what if doctrines required experience for them to activate or level-up?
For example, a country can research doctrines, but those doctrines need experience to flush out errors, recognize nuances, and become effective. If a doctrine had three levels to it, then when a doctrine is first researched, it is level 1 and least effective.
Level 2 could be reached in two ways. First is through doctrine experience gained from combat. This doctrine experience would be automated, not requiring a distraction to the player, much like a general's experience bar. Second, is from observation. What is observation? It is doctrine experience, but gained by observing foreign nations at war that are using the same doctrines you have researched to level 1. In effect, the warring foreign nation is spilling doctrine experience to the rest of the world, albeit at a slower pace than itself is gaining it through direct combat. This observation gives other nations with the same doctrines researched some percentage (10% to 20%?) of the doctrine experience the warring nation is getting, until level 2 is reached. Observation would not be helpful after level 2. I would suggest making observation automated, because militaries observing other militaries has been the case for centuries and does not require a distraction for the player in the game.
Level 3 could only be reached through combat experience. This would create a realistic difference between nations that have combat experience and those who are at peace.
This leveling of doctrines might model history better. Even late arriving nations to the war would have a strong understanding of WW2 warfare if they made the effort to research the doctrines, because the practice and results of those doctrines would be evident, to some degree (level 2) as the war rages. Nations at peace, like the US, could mimic certain doctrines, even introduce their own spin on it, but those who had actual combat experience, would maintain a significant edge in even fights; until the peaceful nation has gained war experience and level 3 in their doctrines. This is different than division level experience or general experience. This could add experience levels and depth to a nation's military theory and its practical application.