So I have reached level 100 for smithing, and I have created dragon armor and raised it to legendary. I did this by filling out the right side of the smithing tree (orcish, dwarven, ebony, etc). Will I gain anything by filling out the left side (plate, glass, etc)?
Apart from the ability to make and doubly improve the armor types associated with those perks, no.
There really isn't much benefit to getting the light armor smithing perks, as the heavy armor branch unlocks the best weapons and heavy armor (Daedric), and both brances unlock the best light armor, Dragonscale. Smithing materials into light armor and weapons could be useful for making money, but, by the time you can make Daedric, money is no longer a concern.