I have a friend who is allergic to potatoes.
Even being in the same room with a potato product will cause swelling, breathing difficulties, skin rashes, the whole nine yards. Just breathing the air of a room with anything potato based in it - so someone opening a packet of crisps in her vicinity - and she's potentially reaching for the epi-pen (which is not a safe option for her, because she has multiple health issues).
Substitute 'Nut' for 'potato' and imagine this policy applied to that. No potato based or potato containing anything in the building at all - in a primary school.
It's easy to say, yes, but no-one needs to eat nuts, but for vegans and those with dairy or gluten issues and allergies, nuts form a large part of the substitutes. And it remains the case that what is applied to one should be applied to all, so:
Nuts
Milk
Fish
Potatoes
Tomatoes
Gelatine
Pork
Beef
Gluten (so wheat)
Those are all in the circle of people I know, all to a serious, time in hospital, call an ambulance level.
Vegan/vegetarian/religious diets, and I don't know what's left to feed kids. Plain pasta, maybe?
Allergy aware is better than allergy ban, and the school should be actively managing it rather than left-shouldering to parents and then wringing their hands saying, 'but we told them!' when it all goes wrong.