There is a scene in my daughter’s favourite cartoon, The Amazing World of Gumball, where the two main characters are in their school canteen, eating food that is completely fluorescent – even by their usual US high school standards. “Dude, I think they put so many additives in my food, they forgot to add the actual food,” one says, before taking a bite and proceeding to turn a radioactive green. My daughter, like most other seven-year-olds I know, finds this side-splittingly funny. From a parent’s viewpoint, it’s less amusing.

A photo gallery of school lunches from around the world a couple of years ago revealed what many of us know, anecdotally, to be true. While Brazil offers rice and beans and fried plantain with a green burst of fresh leaves, the US and Britain stand apart, offering processed meat, too much carbohydrate and too little or even no fresh fruit and vegetables, with tinned fruit or baked beans instead.

So when Prue Leith suggests, as she has this week, that banning packed lunches could be the answer to childhood obesity – on the basis that parents cave into pressure to fill them with unhealthy foods – I can’t help questioning the assumption that school lunches are much better.

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Our problems with food in Britain go so much deeper than the pronouncements of celebrity chefs that I wonder if our diet needs its own #MeToo movement. On the one hand, we’re a nation of plenty; on the other, we have such a reliance on school meals that food banks are having to fill the gap during the summer holidays.

But this is not just about poverty. Food has also come to be about identity, class, race and gender in a way that it is not in other countries. It is why Jamie Oliver aroused the anger of the Jamaican diaspora when he branded a microwaveable supermarket dish “jerk rice”, when it has neither the seasoning nor the style of cooking that legitimise that name.

For the first few years of my daughter’s life, we lived in Ghana, where she ate the same food as us – stews rich in fresh, local vegetables seasoned with small quantities of fish or meat; starchy dumplings made out of corn, plantain or cassava; plentiful fresh mangos, watermelons and papaya. This was not the diet of the privileged either, but standard fare across much of the country. It took less than six months of being in Britain for her to learn what “children’s food” is – processed sausages, deep-fried chips and desserts that come in squeeze tubes, wrappers or plastic tubs. Schools, birthday parties, and children’s menus at restaurants and cafes have been pushing the same messages since the 1980s.

Even when I was growing up, the cult of McDonald’s chicken nuggets was so strong that we actually celebrated our birthday parties in the fast-food chain’s restaurant, the ultimate treat being a tour of the deep freezer.

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At the same time – and in a reversal of what Leith is advocating – my mother took me off school meals and on to packed lunches, because the former was the cause of my childhood weight gain. I had been rounding off my greasy lasagne, tinned sweetcorn and Arctic Roll with custard with a Gold Bar and a KitKat, at 5p each. Access to good food is regarded as a class issue in Britain, but our cultural attachment to junk food transcends the class divide too.

Leith cites Finland as a better model, and with good reason. Finnish children have three hour-long cooking and home economics classes each week. Sweden has pioneered what it calls “pedagogic meals” where teachers and pupils share communal, sociable meals together where they are conditioned into healthy food behaviour and intergenerational eating.

I read about these Nordic countries and, probably like many other British parents, make a note to self to eat dinner more often with my daughter and stop giving her the “kids’ foods” for which she has now developed a very robust appetite.

But life is not how I thought it would be before having a family. You won’t find me or any of the other working parents I know baking bread, experimenting with new, seasonal vegetables or slow-cooking cheaper cuts of meat – as Jamie Oliver says low-income parents should do to get round the problem of healthier food being expensive – during the middle of a busy week. Yet we parents do have agency over what we feed our children. How can we not?

Things are equally bleak at the other extreme of the income spectrum. The investment bank Goldman Sachs has started sending breast milk to hungry infants by courier so mothers who work for the company can be away on business, when I suspect many would still like to be on maternity leave. Sharing meals with our children is simply not something that is valued in our culture. Overworked parents are not the problem, and celebrity chefs are not the answer. We have a lot more work to do than that.

• Afua Hirsch is a Guardian columnist