Actually, it's shown many times with Roku, Aang and Korra that a fully realized Avatar capable of going into the Avatar state at will is only possible after mastering (or at least being competent with) all four elements. Then, the Avatars go from being above average benders who can do the neat trick of bending four elements to being incredibly powerful forces of nature. Aang is never shown to be capable of going into the avatar state at will until he became a competent firebender. Korra also is never shown going into the avatar state at will until after she figures out how to airbend. Roku used to be Sozin's sparring partner before he was identified as the Avatar and sent off for training. After being trained, when he and Sozin have their big blowup disagreement, he is more than a match for Sozin, and levels his entire palace.
It's also about being the bridge between all four nations and the spirits. The avatar is more likely to see things from the perspective of a nation they're not born into if they travel there and spend years mastering the national bending arts. Of course, in Korra's time, this is less important, but it still matters to some extent.
Also, each element is incredibly useful and powerful all on it's own. Think about how powerful and amazing Azula, Toph, Katara, Iroh, Iroh II, Lin, Suyin and Tenzin are, and the feats they have pulled off over the course of the two series. And that's just with one element at each of their disposal. An avatar who can only use three of these elements is basically fighting without a quarter of their full skill set. Terrain and circumstances also matter, if an avatar can't waterbend, and they're fighting in the middle of an ocean, that's a definite disadvantage. If an avatar needs to get somewhere really quickly, or escape their enemies, but can't airbend, that's a definite disadvantage.