van Benthem, in his paper "Syntactic aspects of modal incompleteness theorems" (1979), presented a way of thinking about Kripke-incomplete normal modal logics. He pointed out that for a modal formula $A$ the normal modal logic $\mathbf{KA}$ is Kripke-complete iff $\mathbf{KA}$ is conservative over second order logic with respect to modal formulas, i.e., for every modal formula $B$ we have $\overline{\mathrm{ST}}_x(A) \models \overline{\mathrm{ST}}_x(B) \implies A \vdash_{\mathbf K} B$, where $\models$ is the consequence relation between second-order sentences, and $\overline{\mathrm{ST}}$ stands for the closed standard translation, i.e., the formula without any second-order free variable obtained by attaching to the standard translation the universal quantifiers in the front for the unary predicates that occurs in it.
This view, along with the abundance of Kripke-incomplete logics, made me wonder about the strength of $\vdash_{\mathbf K}$ as a fragment of second-order consequence relation $\models$. Can one say that $\vdash_{\mathbf K}$ is the largest natural fragment of $\models$, where naturality is understood as effectivity, compatibility with the syntax of modal logic, and so forth? Even if it is not the largest, but can one say that it has a peculiar property that is not shared with other fragment of $\models$? Or is $\vdash_{\mathbf K}$ is studied mainly because of historical reasons?