My child is gluten-free, and before we realized gluten was the source of her health issues, bread products were what she craved the most. She is also a picky eater, and I still don't understand why she was craving the foods that were hurting her.
My thoughts about an elimination diet is that it typically only takes a week to see results, there are no harmful side effects from medication, and it's much less expensive than going to doctor visits and running medical tests. Few doctors will tell you to do an elimination diet; you have to decide for yourself. If we had not tried an elimination diet, it would have taken taken us years to figure it out. The potential health benefit outweighs the short-term inconvenience.