We are having to work very hard on DS' diet at similar age. We have been trying for years but it is finally working because DS himself now understands that he needs to work with us on this. He better understands the importance of limiting treats, eating decent portion sizes, exercising etc.
So, what we are doing includes:
He has a small treat allowance daily, based loosely on Slimming World syns. In his case, he chooses this to be in the form of a plain digestive twice daily and a "hot chocolate" (100ml milk, 2 flat teaspoons hot chocolate and topped up with water). All other treats are banned. If he does have an additional treat, the biscuit and/or hot chocolate are cancelled. He makes this decision for himself.
We try to limit bread to a lunchtime sandwich. He has bran flakes with bananas for breakfast most days rather than toast etc.
Portion size and proportions are absolutely key. This is especially true for things like pasta etc because no one needs huge portions of carbs unless they're doing lots and lots of exercise. We eat vegetarian and fish more than half the time. Things like spaghetti bolognaise are limited, not because they're bad, but because it's harder to get the balance of pasta/non-pasta right and they're generally higher in calories and fat as a meal. When we do eat meat, it's often things like steak with some new potatoes and a big salad (oh, and he loves salad because we make them fairly interesting with avocado, pine nuts, nice dressings etc).
Movement is as important as exercise. If he doesn't like getting out and about with you as much (DS is the same), what about playdates in the park with friends? Where they're just running around and keeping moving the whole time (rather than at home where they land up playing electronics or whatever). Or trips to swimming pool etc (obviously, these are also all linked to lockdown easing). Finding a sport he likes is also helpful as at this age they start to be longer sessions which I think is incredibly healthy. Football, rugby, martial arts, parkour/gymnastics, swimming, hockey, tennis etc are all good options. I'm hoping to sign DS up to a swimming club after lockdown rather than just relying on his swimming lessons.
We have really pushed him to eat more fruit when he wants a snack. Often with fat free plain yoghurt.