Are carrots really necessary?

We have recently begun our journey into the wide-world of vegetables. Our first journey began with carrots, however not with the enthusiasm that I was hoping for. Our first attempt was a flop with all of them rejecting them outright. Our second & third try were a little more successful with Everett eating his entire serving. Avery eats 4-5 spoonfuls but Grace (Little Miss Princess) has absolutely no desire to give them a fair try. At least we know her gag-reflex works. She is VERY similar to Auntie Shannon when it comes to food. Grace will store everything on her tongue, and not swallow. We even try to trick her by serving cereal along with the carrots, but she was quick to catch on to us, and even refused to eat the cereal.
We then moved on to the sweet potato, and I thought for sure they would all be very interested. Everett ate his entire bowl (we are now up to 1 tbsp), Avery 4-5 spoonfuls, but that girl Grace, gags at even the smallest spoonful. We have a few more days on the sweet potato then we are moving to peas. Grace will be just thrilled.