I find it useful to think of systems that use proofs as "geometries" of more or less metaphorical "spaces." So, for example, Euclid's geometry defines a plane, game theory maps to "decision space," and logic itself describes what might be called "logical space."

In a plane, there can be only one line parallel to a second line through a point outside that second line. If that condition is not true of a space, then the space is not a plane, and deductions about it won't map to real-world two-dimensional phenomena.

In game theory, the optimal strategy is the one with the greatest pay-off. That is the nature of a game space. If the object of the decision is not to optimize the outcome (by the player's lights), then game theory does not describe the decision to be made.

Geometry and game theory use mathematical logic, i.e., they inhabit a "space" in which logical inferences hold. A system of inferences in which conclusions do not follow from premises does not map to that space, i.e., such a system is "illogical." So, yes, the rule that conclusions follow from premises is "just a rule," but it can be seen as THE rule that DEFINES logic itself in the way other systems' axioms define the spaces those systems exist to analyze.