Update:
According to a thread on a well known Xbox website, I am not the only one with problems. So I am sure now, that the game has some issues. The page where I have found a thread is called TrueAchievements and their exists a sections where you can report glitches and bugs

I think the game data ...

Hi,

I have problems with unlocking the Achievement "Blitzkrieg". The requirements are: "Complete the Blitzkrieg France campaign."
I have done that, and even replayed the last mission completely (by restarting the whole level with the "back" button by using a previous save file). But each time ...