old rivalry
Hard to know who that rivalry is between unless you mean France and Germany which is somewhat asymmetrical given that you've conquered them three times (not 3x within living memory nowadays though I'm old enough to remember people who were over 90 when I was a kid who were alive for all 3 invasions).
Generally, it's petulance from the French who are sore losers and still like to think they're a superpower (which they've not been since 1815) and the navel of the world.
There's no reason for a European Army. Europe isn't a country, isn't a unit or a union or an alliance, however we do know how to cooperate when necessary. Our strengths are in our differences. French bloody-mindedness and German inflexible bureaucracy would not enhance the status quo and the closest example of pan-European cooperation which involves most but not all European powers is the EU which is far from being a shining example of flexibility, creativity or efficiency.