People like to complicate things until they are no longer accountable for themselves or their children. That people today wonder why a 4 year old is overweight is part of that dumbed down brainwashed mindset.
You say calories/activity is only an indirect contributing factor for a 4 year old becoming very overweight, I say it is by far the most likely reason and by far the most direct reason, still, even in our polluted lives. Earlier I gave reasons for diet quality influencing weight gain (as you did soda), but diet volume and activity still make those raw materials available or not to be manipulated by the drugs/soda. Spin it any way, it still is based on in/out raw materials which as you say can be *influenced* by other factors. The 4 year old? I guarantee as much as I can over the net, that the kid needs less food (and better food we agree on) and more activity.
I don't want to get trapped in semantics here. All of your examples may *influence* weight gain. If everyone today was placed in a concentration slave camp, they would all lose weight, correct? According to your logic the ones with "weight conditions" would continue gaining weight as the others withered away. Again, you can not gain weight unless the raw materials are there by extra consumption an/or availability because it's not being burned. Diet soda has no calories so it can not directly cause weight gain - this is science. I don't consider water retention/bloating to be weight gain in a "very overweight kid" scenario.
I think most people are non-accountable for their own health because they have so many conditions to blame instead of getting off their asses and stop eating junk and learning about nutrition. People want to blame the soda or the genes instead of taking accountability for themselves and their duty to raise their children correctly. I think soda is evil but I do not believe it directly causes weight gain. Anyone drinking diet soda is already nutritionally ignorant and this likely translates into other bad diet choices which do cause weight gain.
So, in short, get that kid out of the house, out of the doctor's offices, away from the tv and computer... and give him a better diet. Sunlight regulates hormones which indirectly affects mood sleep and hunger cravings, which would help but input/output still regulates a person's overweight status.
Reminds me of the 800 pound guy who was stuck in his bed for years while people brought him food and enabled him to become "very overweight". Then they blamed it on some condition instead of blaming him and the people who stuffed his mouth while he laid in bed watching tv for 5 years.