I have schlepped what seems like entire grocery stores on flights for my soon-to-be 4 year old son (about 12 flights so far, and another one on monday). In fact, when we traveled to Italy last year I had a humongous bag dedicated entirely to snacks! It weighed a ton but I had to make sure to have enough for both 10-hour plane rides and our week-long stay in three cities because I didn't want the hassle of having to go to the market for "kid stuff" there.
That said, I have been surprised at what they let me bring. I've never had a food turned down (including juice boxes, single-serve milk boxes, applesauce cups and yogurt squeezes - they are unopened and TSA does a swab test on them). Any other non-liquid food was carried in any condition, e.g. didn't have to be original or sealed packaging.
The only time I purchase "gummy snacks" for my son is for flights so he knows them as plane treats and I serve them to him during take-off so he'll be chewing something. Other than that we have the standard issue fig newtons, crackers, twizzlers, granola bars, chocolate soymilk; basic portable snacky stuff that's not too messy or involved.
One thing that surprised me when I flew with him two months ago was being allowed to take ice through security. My son had finished his drink but we have a pretty great thermos that kept the ice so cold it didn't melt and I didn't realize it until we were almost to the scanner. I explained my oversight and asked the TSA agent if I could dump it out to send the thermos through. He opened it, checked that it was just ice and said go ahead. Apparently, explosives don't freeze so ice is fine.
Oh and ALWAYS have an emergency lollipop stashed that they don't know about! On a flight from Philadelphia this summer I didn't need it for my son but offered it to the parents of a late 2/early 3 year old who was having a meltdown. Problem solved.