DMing has helped me with speech for years. Years. Not even joking.
Think of it: You have to take the lead in a massive conversational thread. You have to “keep it interesting.” You (eventually) unselfconsciously “do” voices. The story you tell is one that must interest you to successfully carry your message. You keep the story linear (enough) to be sure you are understood. To keep Players’ attention, you have to speak clearly, concisely, and resonantly so all hear and comprehend without your having to repeat everything you say.
Where else does D&D prove useful?
Math. Players must calculate damage rolls all the time. As DM, you do that about three times as much during the course of every session – damage rolls, encounters, chance, and even just to freak out your Players into thinking you’re rolling for a reason. (This time, right? This time?)
Writing. Not only useful for stringing together plausible and coherent thought, you can build novels out of the stuff games are made of. (R.A.Salvatore certainly did this, though – I think – he created Drow. Or Drow society, at least.)
Music. Maybe even here, yeah. Theater people, those who play Bards. Why not sing during gameplay? So long as you aren’t shattering glass (or others’ eardrums), it lends verisimilitude and deeper in-character traits.
In short, don’t limit the ways in which you can expand natural talent for storytelling or for skill/intellectual improvement into other areas of your life. Makes you smarter.