Germany has already legalized allowing EU citizens to join the Bundeswehr.
And a EU military has been planned literally for decades, including how to unify all the militaries under one command structure. This military will not defend us in the same manner as we defended them for decades; it’s not in their remit.
It is useful to note that NATO was not formed at our initiative, but at the urgent pleadings of post war Western European countries. They were facing hundreds of divisions of the Red Army salivating at the thought of racing to the western coast and absorbing the rest of flattened fortress Europa. The set up was a famous "long telegram " from ambassador George Kennan in Moscow who advocated for the policy of containment of the Soviet Union in 1946. NATO was singed into law in 1949 by the western democracies.
With the collapse of international soviet communism, and the dissolution of the Warsaw pact, the reason for NATO's existence vanished. It exists now only because of the natural law that says all bureaucracies find a way to perpetuate themselves.
There is no reason why this 70 year old institution should not be re-examined in light of major changes in the world at large. After all, NATO is not a church. It can be replaced by some updated alternative.
The Europeans are big boys and can take care of themselves if they so choose. EU has three to four times the population of Russia, and is far more wealthy. France and Germany love to throw their weight around in the EU-- let them muscle up and take on their own defense along with their neighbors.
This does not mean we shouldn't have a mutual alliance with the EU, but it doesn't have to be the old NATO framework, which in truth mostly meant the US military on European soil.