Rhythm is the spirit of jazz
Rhythm is the spirit of jazz, its source of vitality.  Any musician who plays this music is in some ways a drummer.  Listen to how the rhythmic sense of this incredible trio brings the music to life!  Each of the individual musicians in this group, Jim Hall, Wayne Shorter, and Michel Petrucciani, has the developed rhythmic sensibility of a drummer. 

So much rhythm!
Melodic instrumentalists as inspiration for drummers 
I have had some spirited discussions with musicians on the AllAboutJazz forum about the value of learning the drums for any instrumentalist.  But the reverse is also true, that is there is a great value to drummers in studying melodic instruments.  For example listening to the music above, Michel's solo starting at 3:48 is full of incredibly sophisticated rhythmic phrases that are played with real finesse.  Another obvious example would be Jim Hall’s intense rhythm guitar groove throughout this song, any drummer would be proud to lay down such deep sixteenth notes. 
Drummer tunnel vision
Because of the fact that anyone can play like a drummer, drummers can and should find inspiration from many different instruments.  There is a long history in jazz of musicians studying other instruments, and then applying the lessons that they learn to their original instruments with great success. A classic example of this fertile, pan-instrumental inspiration are Lois Armstrong inventing scatting by singing like a trumpet. 
You dig?
This doesn’t mean that I think that drummers should stop practicing their instrument and practice something else instead, or that they should stop studying all the great drummers.  What I do believe is that the world of jazz is too large to ignore everything that isn’t the drums, and that the more you learn about other instruments, the more well-rounded and musical you’re drumming will be.