Coincidentally, what made him into a healthy eater is that he found that food with lots of sugar and fat made him feel worse. Oddly, it doesn't happen to me at all, but he grew to be sensitive to it for some reason that we never understood. But he eats extremely healthy, if not a bit bland, now -- and is a great athlete.
The evidence will come later if it hasn't already. Their behavior might be a hint. It impacts focus, brain development, everything.
Well, I can only conclude either you don't have kids, live in a very different setting, or you're some kind of super parent :-)
The main reasons are (a) the kids are highly attracted to foods that constitute a poor diet and (b) they are constantly exposed to opportunities to consume those foods.
At its peak we were taking them to 2 - 3 birthday parties a weekend at which they were presented with unlimited candy, giant pieces of cake, sugary drinks, and a humungous bag of lollies to take home. Then at school the same children bring sweets to school and hand them out to the whole class, trade things out of their lunch boxes at lunch time, etc etc. Unsurprisingly they routinely come home having eaten only a small portion of the food we gave them and tell us they "weren't very hungry".
One of the interesting revelations as a parent has been how little control you really have over what your children are exposed to and how much is dictated by the society around them.
Notwithstanding the food-swaps at school (I think to liken this all to be candy is hyperbolic), the kids eat what you feed them for breakfast and dinner. Whether this is a battle has little to do with the nature of the food itself. I'm sure you don't believe that perpetually relenting to kids when they whine, piss and moan will make your life easier - they'll normalize the behavior to get what they want and misbehavior will be the norm. Being a parent is the battle and it doesn't end at letting kids eat candy any time they want, which makes healthy eating a moot point.
Another thing I didn't expect is arguing with the spouse about it, who will also undermine the effort.
One good thing my parents taught me is that vegetables can be eaten without cheese or salt, and I currently eat healthier than anyone I know, and it's not a struggle at all for me.
It's just "normal" and I'm accustomed to it. I think this can still be trained as an adult, but it's probably easiest if you never have to "transition" or break out of bad habits. Wishing your family the best.
It's the same with all things in parenting: you do your best and hope they will understand you did it out of love. But there are no guarantees.
Young bodies can tolerate even crappy diets. It's a different matter when older however. If it isn't turned around by their fifties say, they'll be in for a world of hurt. That's why it habits must start young.