Most Sephardim do say a Mi Sheberach for the soldiers. And there are also a few Ashkenazi communities that do (the more modern ones).

Those who don’t—it’s a byproduct of the battles they fought over enlistment, from which they came away with the feeling that the army is a sort of minor enemy. Saying that outright would make the repressed stupidity too obvious, so they found pseudo-halakhic excuses.

That’s different from Hallel on Independence Day. The motivations are the same, but for not saying Hallel there is a real halakhic case. A Mi Sheberach, in principle, you could make even for your sick dog—though in that example I might agree that it’s inappropriate.

Maybe deep down they recognize Michi’s teaching that the blessing doesn’t really help. If so, then it’s just identification/solidarity, and that’s already not their thing…

By the way, I got from here an argument against “we protect the soldiers through our learning” (I’ve already had quite a few tell me they added another page of Talmud to their daily output and the like).
Oh, studying three full sedarim for the sake of the soldiers you can do, but squeezing out a Mi Sheberach is too hard?