See, and I think the OP is sensitive enough to her child to balance his need to control what he eats in the present and his need to learn good manners and competent self-care. I don't think she's going to force her kid to eat things that make him vomit and I don't think she's going to raise an ungrateful, ill-mannered person.
There are a lot of things to keep in mind when we address this issue. Do you want to have a big conflict with your kids about food? My answer: not even in the short term, your answer may be different. Do you feel put-upon if you have to cook different things for every member of the family? My answer is no, but many parents really hate doing this. Are you worried about allergies? Me: no, you, maybe yes.
Do you value raising your children to seem French, and are you willing to cook your way through Julia Child in order to make that happen? I don't want a French kid, no matter how much fun it would be to cook like that...I know that's not the point of the book, but I'm getting a little tired of trying to make my kid into a Parisian or an Amazonian hunter-gatherer. I want to raise him in my culture, thanks. For heaven's sake, I actually WANT him to be picky and not to eat foods he doesn't like. Seriously. It was not good for me that I ate whatever people asked me to eat when I was a child. He needs to learn to assert himself politely at the table without hurting the cook's feelings. That's also a skill.
I think you have to look at yourself, and look at your kids, and decide which feels like the right way to you. I've seen both directions done badly and done well. It's really no fun to have a gathering with parents who expect children to eat everything if their kids don't want to do that. Yes, that can sometimes work but sometimes, it's just a lot of parent-kid bickering. Ick.