Even though it is realistic for fighters to be trained in multiple weapons, particularly if they were in an army, characters in fantasy who use weapons almost always have a specific weapon with which they specialize, and thus almost ALWAYS use in battle. I think 4e does this fine, as a fighter can use any military-grade weapon if they have to, but can choose feats to specialize in using their favorite weapon better. 5e only has a few feats so far, so we only have a rough idea of where they're going with weapon specializations, but it seems that it is largely the same: the fighter has training in every weapon category, but if you choose, for example, the Archer specialty, you are better with bows than other weapons.