My wife and I have worked hard to feed our 2-year-old daughter whole food, to keep sugar and processed crap out of the house. We try and do dinner together as a family every night (not always possible), and we always try to give her the same meal - or close to it - that we are eating.
I don't know what other kids eat/don't eat, but our daughter loves mushrooms, onions, peppers, carrots, peas, broccoli, lima beans, berries, banana, avocado, sweet potato, beef, chicken, turkey, shrimp, and eggs. She does some white rice (and loves going out for sushi). She still does some dairy - whole milk and high-fat cottage cheese and hard cheese, but we try to keep that in check. She also drinks water, eats some raisins and nuts for snacks, and, in our one concession to making our lives a little easier, organic, no-sugar-added applesauce.
We don't do snack crackers, cookies, cereal, pasta, cakes, juice of any kind, or other processed foods.
She seems to be thriving, is happy and healthy, and in our opinion and good and adventurous eater. Nothing has made us chuckle more than watching her eat about a dozen pearl onions cooked in butter like they were popcorn.
I think it really starts and ends with the parents. If you, as the parents, are eating those things, and that's what's in the house, then that is what the kid eats, and what the kid assumes is "food." I know we won't be able to keep all the other stuff at bay for ever, but I feel like every month we can hang on without it in our lives, the better off she'll be.
“If you want to build a ship, don't drum up people together to collect wood and don't assign them tasks and work, but rather teach them to long for the endless immensity of the sea” -- Antoine de Saint-Exupery