Different modal verbs have different meanings. Otherwise, there would be no need for them in our languages, and no need for alternate modal logics.
Necessity implies truth in the traditional interpretation of alethic modal logic because something is necessary if it is true in every extension of the current reality, and the current reality is an extension of itself. (We tend to honor this interpretation because it fits normal usage and goes back to Aristotle.)
That does not mean this is the case for every version of the box and diamond operators, but if you are using a modal logic where the box does not imply truth, most folks would say that is not a model of necessity. There are at least three modes in basic English grammar alone, many interpretations of each, and many other kinds of modal logic unrelated to them.
Clearly, modes like "should/may", capturing obligation vs permission or 'would/might', capturing predictions and options, do not require either operator to imply truth, because I can shirk responsibilities and avoid or create circumstances. The fact that I am obligated to do X does not imply that I will do X, it implies that if I do not do X, I am not compliant, and suffer whatever the loss is for non-compliance (perhaps nothing). Likewise, the fact that I would do X in some extension of reality only matters if reality becomes something like that extension. More specific models of belief, certainty, awareness, etc. refine these basic modes, and are still topics of modal logic.
