I wish I could give a comprehensive answer to your question, but my knowledge is limited. I'm moved to reply because I think the comments to the question are giving the OP a hard time; I think the question makes excellent sense. Unusually, I disagree with Mauro: logic is not fundamentally about truth, it is about consequence, or deducibility. One of the striking things about the relation of logical consequence is that it is not confined to statements with truth values. Many, perhaps most, speech acts possess their own logic. Obligations can entail other obligations, commands can entail commands, necessities can entail necessities, etc. Validity is not exclusively about preservation of truth. A valid argument in a deontic logic is one that preserves obligation from premises to conclusion. An intuitionistically valid argument is one that preserves constructive provability from premises to conclusion.
Typically, it is a standard move when dealing with such modalities to introduce box/diamond notation and to replace "A is necessary" with "□A is true", for example. We commonly wrap up semantic properties and put them in the box in order to reduce everything down to truth. This tendency towards a semantic monism in which only truth is considered important is fairly pervasive, but I share the OP's desire for it to be justified. After all, many theories of meta-ethics do not assign truth values to moral judgements, so why should we assume that a logic governing moral obligations should be expressed in a truth-theoretic way?
An initial observation might be that since we already have a calculus that expresses the implicational behaviour of propositions that are connected by 'and', 'or', 'not', etc, it might be a duplication of effort to create new rules to show how these connectives behave when they connect things other than propositions. The axiomatization of a modal logic has the effect of demonstrating how to reduce sentences involving connectives between modalities to sentences involving only propositional connectives. For example, the K axiom of modal logic effectively reduces strict implication to material implication. If we can axiomatize a modal logic and perform this reduction correctly, then this suggests that the logics of modalities other than truth are eliminable.
Another point is that we desire our logical operations to be computable. We might argue on the basis of the Curry-Howard correspondence that our best understanding of computability corresponds to the concept of provability afforded by classical logic. And classical logic has the natural semantics of truth and falsehood. This suggests that any logic worthy of the name is ultimately either about truths and falsehoods or about something that looks just like them.
Another consideration is that logic has epistemological consequences. We use logic because we want to come to know things by inference from other things we know. But we usually conceive epistemology as being concerned with knowing things that are true. Think of all those theories of knowledge that are taught in epistemology classes: "X knows that P iff..." and typically one of the conditions is that P is true. Maybe such accounts are too limited and we can have knowledge that is not of things that are true or false, but detractors might claim that we would be describing mere sensibilities and not real knowledge.
But all this still leaves the question that if the logic of modalities is reducible to that of truths, what are modal claims true of, exactly? Are there really necessary truths, moral truths, even aesthetic truths, etc? Modal logic is commonly expressed using Kripkean possible world semantics, but does this mean that if we judge a modal claim to be true, we are committed to the existence of PWs? David Lewis thought so and embraced modal realism, but his position has not proved popular. Others have adopted anti-realist or quasi-realist positions. I suspect that in the last analysis a full answer to your question can only be given on the basis of a comprehensive account of truth and realism and the relationship between them.