I recently had a conversation with some other musicians, and this question came up:
“Does studying and playing another instrument make you a better drummer?” 
My response was “yes.” I think piano, especially helps teach you some musical fundamentals such as keys, chords, scales, phrasing and even some theory. Additionally, when you play drums and understand the way the other musicians are thinking as they build to the bridge in a tune, or pause (without any drum flourishes) to let the tune breathe; then it just makes more sense to you. It also makes you more aware of when to……and when not to……play a drum fill.

If you are going to write songs, then piano (and often guitar) are good instruments to help frame the structure of the tune and demonstrate your idea to others.
Is it mandatory to play some melodic instrument in order to become a good drummer? “No, it isn’t.” But there is no question that an enhanced understanding of melody and other instruments benefits your total musical knowledge. Maybe that’s one more reason to be a well-rounded musician, not just a musican who only plays drums.