"Gamey?"... a good description. That flag denotes the military control of the area, not the political ownership and it is temporary only. After Russia has retreated back east, the territory will convert back to its political ownership.
A good example of how this bears of game play is as the French when you invade Russia. As your corps pass through a territory, the territory may change its control to French with the French flag. not only is supply now running through that area but additional forces will move more quickly through the territory. However, if you pass through a territory too quickly, the military control may not change and you will not yet be able to pass supply through that area.
A good technique to speed the rate of invasion is to send cavalry divisions up your route so as to seize control and therefore speed the rate which your army corps will pass through them. The setting of your cavalry will matter since the more passive the mode, the longer it takes for them to convert the area to their control. Use the "2" button to change the map shading to show the control and supply lines.
The technique of proceeding your army corps with a blanket of cavalry is also important in 1805 when you have a race to Vienna. If you take too much time to get there, the Russian Army will have arrived and the battle of Vienna will be much more costly.
Using cavalry in this manner also acts to screen the advance of your infantry, as it was done historically. However, you need to be prudent doing so, otherwise your cavalry divisions can be bumping into the retreating enemy's rear guard and getting a "bloody" nose. Cavalry strength afterall is expensive to replace.
And this supports Napleon's Military Maxims. ... that territorial objectives are always secondary to the destruction of the enemy army and the preservation of your own.