Shutup I feel for you and your DC. I think you're thinking in the long-term and you're right to. My nephew, who spends a lot of time with me, had a similar problem. Could I offer a couple of suggestions?
I would basically cut out the snacks, except for one snack in the afternoon about 3-4pm just to tide them over til dinner. Kids don't need to be shoving food in their gobs all the time, even healthy food. They should be genuinely hungry when they sit down to dinner. I let my DCs start with a veg course - a small salad, or a bowl of peas or broad beans that she shells, or some cherry tomatoes, carrots whatever. Then give them a reasonable portion (not small, not enormous) of tasty food - whether it be pasta, fish or meat based dish. I think you're absolutely correct in not wanting to make an issue out of this, and you don't want to go down the "forbidden food" road, or do calorie counting. It's not right at this age.
Also, and this was just something I noticed with my nephew... does he eat really fast? Like not tasting, just shovelling? Dnephew ate about three times as fast as anyone else. I think physiologically he simply didn't feel satisfied because their wasn't time for the brain to register this - does that make sense? Anyway, we spent time encouraging him to slow down, put his fork down, make conversation, relax, and it really made a difference. He tastes his food now, and eats less because he's satisfied with less.
Finally, get your DC walking. To the shops, around the block, to the next bus stop etc. It all adds up on the energy burning, and for a "lazy" kid, this might be the best option to start with.
Good luck, OP!