Well, that's the gig; lots of other games have some pretty easy ways to have your cake and eat it too, or don't actually tie up most of the basic character design with combat. An old-school typical RuneQuest character would probably want to maintain whatever one-handed melee weapon and shield he used, probably a ranged weapon, and give some attention to a backup weapon like a shortsword or a dagger. But that left plenty of training time for Oratory, Persuade, his Perception skills and the like if he wanted to do so.
With the D&D sphere (outside of, to some degree 13th Age since Backgrounds are disconnected from class), you really want to take one of the classes that support skill use if you want to have it good, and those are usually not as direct combatants (though some, as you noted, do a lot of valuable combat support depending on class and the specific system).