Banning foods that contain any amount of added sugar, except for sweets that are explicitly marketed as such, is very reasonable and would go a long long way in helping manage the obesity crisis.
For instance white bread has a high glycemic index not because of added sugar, but because all of the fibre and most of the protein have been removed. Proper whole grain, brown bread has a low glycemic index and tons of protein. And that's true even if there's some syrup added for flavour, which is not uncommon.
The economic pressures are at odds because as people eat less bread shelf stability, both at the grocery store and at home, becomes increasingly important to maintaining a desirable yet competitively priced product. People don't want added sugar and are more carb conscious generally, but what they dislike even more, without realizing it, is stale bread. So you get a positive feedback loop that turns people off of bread, I think.
AIUI this is because the whole wheat flour tends to impart a bitter taste, so the sugar balances it. The sugar also enables the yeast to raise the bread faster, helping mass production.
Having said that, I see your point that setting too low a threshold might be unnecessarily onerous on cafeterias and manufacturers. Perhaps a threshold of ~1g sugar per slice would be more reasonable?
That's not nothing of course, especially for a school trying to minimize food costs. Many schools (like the one I attended growing up) procure the cheapest possible USDA/NSLP-approved product.
But kids at these sorts of schools seem like they'd see the greatest benefit from this sort of legislation to raise the threshold of what constitutes a healthy lunch.