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I think there might have been some nuances to @MikeTheBear 's answer that were missed...No, it's not!
Kettlebells and/or calisthenics can get you to "strong enough"***. You don't need barbells for "strong enough", hence they are definitely not necessary.
But like I said in my earlier post, they are the best tool for absolut strength and provide the fastest road to get there.
So no they are definitely not needed to be "strong enough for military purposes", but most likely the fastest and most time efficient tool to get there.
Have a look at Tactical Barbell I+II for a good routine that will make you "strong enough" with the needed endurance for military or LEO requirements.
***we throw around the term "strong enough" a lot lately. "Strong enough" of courses varies. For LEO personal (I don't know if it's the same for military) there are different "strong enough's".
For example a breacher certainly needs more max strength than a sniper.