It is reasonably cheap to eat healthily, unless allergies or fussiness come into play.
For my part (bill has increased by around £30/week), I am now back to dinner planning rather than buying on a whim, but other than that our meals are still very healthy and balanced.
We typically have:
-wonky versions of fruit/ veg, unpackaged unless the packaging makes things cheaper (rare, but it happens)
-frozen where it makes sense (e.g. I now only buy frozen onions as most of the onions I buy fresh are rotten) or where portioning means I will have leftovers
-fish, chicken and mince frozen only unless I need the mince for meatballs
-at least once a week cheap organs - heart, liver, kidney - cooked directly or made into pies
-on the rare occasion I buy expensive meat like duck or beef, it is only ever as a side part of a meal, never the main bit (e.g. for stews are stir fries)
-where fresh fruit is expensive, I buy either frozen for berries (zero mould) or tinned for "pudding fruit"
-old fruit is used in cooking (apples) or frozen in protions to make smoothies
-old vegetables go into soups, stews, blended into sauces or chopped into stir fries at least once a week to stop wastage
-pulses are healthy and filling and will stretch most meat dishes nicely - so will mushrooms
-eggs, while going up in price, are still incredibly cheap and filling if they form the base of a meal like omelette, foo young or as part of a homemade sandwich filling
-water is a staple drink; it comes from the tap and as my children believe ice cubes make everything better, we have a few trays of ice cubes on rotation - this also thins juices (which, for the youngest, are always watered down anyway)
This all is very doable if you can cook and know what goes together. Helathy food really isn't that expensive if you're willing to swap and improvise a bit. But people do need to learn how to cook.
My skills came from children's and student cook books as well as asking elderly family members or small, independent eateries for recipes if they had made something particularly good (enough politeness and a smile usually worked well on the latter), so the whole premise of it being down to parents and schools doesn't really come into it. Now is as good a time as any to start.