Is it really impossible to make a protein bar savory?

In order to make something that is shelf stable without refrigeration, it needs to be either hostile to harmful microbes or sealed in a way with no harmful microbes inside (and will have to be refrigerated after opening).

There are a few ways to do it without sealing, including reducing water activity low enough that microbes can't grow. Flour, rice, oats, nuts, and other bulk dry goods generally follow a dehydration process. Oil doesn't have water in it, so sometimes there are high oil substances (peanut butter) that don't have enough water to support microbial activity.

Another way to reduce water activity is to bind the water molecules with other molecules. Sugar is by far the most common substance useful for reducing water activity, because it's possible to mix water with a lot of sugar. Honey is shelf stable because it's something like 15% water and 85% sugar. Maple syrup is about 33% water and 67% sugar. At those sugar levels, microbes struggle to actually resist the osmotic pressure and use the water present in the substance.

Note that salt can't really do the same thing. A brine that is 95% water and 5% salt is basically inedibly salty. But 95% water is still top high to really inhibit microbial growth. At most, you hope that good microbes outcompete bad microbes (this is the basis for pickling sauerkraut, Kim chi, certain types of pickled cucumbers, where lactobacillus strains will outcompete harmful bacteria and mold). But even these foods may keep much longer when refrigerated. Even soy sauce, at 16-20% salt, is recommended to be kept in the refrigerator (for quality, not necessary for food safety).

There are other ways to inhibit microbial growth, or just the harmful microbes: acid or alcohol can do a lot.

But as a result, the easiest way to make a shelf stable bar is to dehydrate it, maybe add a bunch of sugar, and use ingredients that still have good taste/texture when dehydrated. So they use a lot of things like nuts, chocolate (high enough sugar to have low water activity), trapped air bubbles (good crunch when totally dried out). And the sugar allows it all to bind together.

And there are other ways to bake savory goods. They just have to be crispy all throughout, and usually thin enough to bake/fry dry without making it too hard to be pleasant. Think chips, pretzels, even savory mixes like Gardetto's or Chex mix. Even the bread stick components have to be dehydrated to the point of being brittle and crispy, like a crouton. Turning that into a shelf stable bar form that actually tastes good, without adding sugar, would be difficult.