I was going to say, "Go for it," but I see you already bought one for your child.
My almost 3 year old uses my iPhone to play games, listen to music, look at photos & watch videos. It is amazing how well she can navigate through all the stuff on that phone.
We do make a big deal about having her sitting at all times with it (to minimize the chance of it falling & breaking). If I had the extra funds I'd definitely buy one of her own (so I could reclaim that disk space for my own stuff). Mine is only an 8GB, so our entire library isn't on it..I rotate movies, TV shows and some songs. But when my hubby finally upgrades from his old/original iPhone, it will be passed down to the kids without the phone feature.
We also have an "ancient" 5th generation iPod that still works fine, but it is not particularly user friendly for the preschool set. Our 6 yr old can handle it, but navigation is an issue for the 2 yr old.
I just need to add that having & using the iPods doesn't prevent us from having real conversation, playing tic tac toe (both on the phone & on paper), playing I Spy and all sorts of old fashioned fun. Technology doesn't always spoil us, it can actually enhance our lives.