Woodchukhntr:
I too made a sling when I was a kid and was astounded at the power you could impart into a golf ball sized rock.
It all came about when reading a book about the Incas and how they fought the Spanish (to a stand still) using slings - while the Spanish were using blunderbuss.   "A sling-shot?" sez I.   Later on, there was a minimal discussion about the sling they were using.  Grabbing two lengths of rope (one of which was tied around the wrist) and a piece of canvas, I had a sling that would easily chuck a rock 100 yards.   I never got good at accuracy, but I sure could send them downrange.  (The Incas would mass a bunch of sling throwers and through sheer volume, achieved their objective.)
So, is a sling adequate for deer?   I think so.   
(But I think for the time being, I'll stick to my 20 ga Handi.)