I think the 'home cooked food' thing is a red herring (unless you eat constant take-aways). As others have said, you can cook using healthy ingreedients and limiting saturated fat, or you can pile that stuff in (like my MIL does with her 'home-cooked' meals). It also depends on portion size, tummies do expand to fit the food available, I know this myself when I have managed to expand my stomach to eat more, I get fatter!
Children are less active than they used to be. Even with lots of activities, they are just less physically strong than the last generation. There was a study recently that measured this and today's 10 year olds were about 25% less fit than in the 1970's.
I would say that in my daughter's ballet class (which has terribly unforgiving leotards), about 1/3 of the 7 year olds are chubby/fat. That is quite a lot. I don't know if that can be all genetics, even if some people are predisposed to being a bit bigger built.
This sounds awful, but if my children were getting chubby, I would up the exercise (which they do a lot of already) or limit treats/sweets/cut portion sizes down. Not in a drastic way, more a fine tuning. As it is, their mostly healthy with too many treats in it diet seems fine for them. I wouldn't obviously smack their hand like FreePeace (how awful, no wonder lovely biscuits seemed so tempting), but move it towards one biscuit and an apple rather than two biscuits, one ice-lolly instead of an ice-cream too often, tiny changes rather than an all out diet (which doesn't work for adults either).
And, I think carrots are fine IMO as one of the things you can snack on between meals, surely it's a balance (e.g. carrot/piece of toast/handful of nuts/glass of milk/apple) rather than having one thing as your snack. Nuts are great, but lots of nuts every snack-time is going to be fattening.