I'm afraid I do think you're being simplistic. I confess this is from a personal perspective. I never did Domestic Science in school (girls' grammar school) and my mother was an appalling cook, but somehow I learnt to cook reasonably well, healthily and cheaply. I did learn about basic nutrition in biology. There are three macro-nutrients: protein, fat and carbohydrates. Anybody eating food from a variety of sources will not be short of micro-nutrients. We don't actually need to eat carbohydrates (especially sugar), because the body will convert fat and protein to glucose as necessary, but carbohydrates are cheap and (let's face it) taste good.
There is so much information available now for free on the internet for people who really want to know about nutrition and have accepted that they are responsible for what goes in their mouths. Unfortunately, there's TOO MUCH information and thousands of snake oil nutrition salespeople promise magic wands. The basics of good nutrition isn't really that hard to grasp, but that doesn't take into account the psychology of eating, which I think is much of the problem.
If you're not overweight, you've probably got your food balance right. If you're overweight, you need to cut down your calorie intake. You need to be honest with yourself about what you're actually eating (especially portion sizes) and cut down. It makes sense to cut down on empty calories - the ones which are only providing energy (calories) without additional nutrition, so read food labels. Simple!
Obesity is a national issue and concern. Having said that, I get a bit fed up with being 'blamed' for an illness labelled as preventable. I was diagnosed as a Type 2 diabetic in my mid thirties without ever having been anything more than slightly overweight and having been reasonably active. There's something going on in my metabolism which isn't accounted for in the usual explanations of diabetes.
Lifestyle factors are costing the NHS, but they're not the major reason for queues in A & E, etc. I don't think health education should be the responsibility of the NHS.