I do think that solving the food problem is important, because it limits a person's social world to not be able to eat in restaurants, or be invited to dinner and know they might not be able to eat what’s there. It is part of the overall project of raising a person who can be flexible enough to participate joyfully in whatever comes along in his life.
That is COMPLETELY true.
I also think that there are some kids who are simply immovable, too, though. My DD hates vegetables, and it doesn't matter how many ways we tell her that they are: a) safest, b) easiest, and c) the healthiest choices that she can POSSIBLY make with respect to food (and they are-- processed anything is higher risk by far), she'll go after carbs and processed meat/cheese every time instead.
At nearly 14, I don't see us really budging her on this one. I'm waiting until I can dance around her and tell her "I told you so! I told you SO!" when she develops the early signs of scurvy. That's probably not going to be my proudest moment as a mom, but I may do it anyway given all the grief she's given me over the years.