People on diet have lower performance because their brains are lacking fuel on diet.
Absolutely false. When I'm not "on diet" IE eating three square meals full of traditional American junk, I feel like garbage, my brain is fuzzy, and I'm generally rubbish at my job. Cut out the sugar & starch, eat enough protein, and I can easily eat only one or two meals a day (approx 1200Kcal), and work several times better. My brain has plenty of fuel between those limited calories and the lard on my ass that's getting burned off in the process. There's an acclimation period of about a week that's no fun at all, but once over the hump, I'm a much, much, much better code monkey when I lay off the bananas and stick to protein.
As far as being fat/not fat, exercise doesn't make much difference. You can't burn enough calories working out to make up for over eating. Can't out run the fork as they say... Exercise makes me feel better. I'm an endorphin junkie, no question. When I feel good, I stress/depressed eat much less, so that helps in weight management.
"Enough" exercise to get that effect in no way requires me to never see my family. I run three times a week. I do it at lunch time (those would be the one meal days usually), and I'm not actually away from my family any longer than if I'd spent lunch going to Mc HogBurger for a triple artery blocker... Or here's another idea... Want to be a present father? Exercise with your kids, and maybe even their mother too. Turn it into a family affair, teach your kids to enjoy something that's good for them, and be a far better parent than most.
If you want to exercise, you can absolutely find the time. If you don't, you'll spend the time finding excuses instead.
Source: Dropped about 180 lbs over 18 months by eating less and exercising more, all while working full time (plus standard tech career unpaid overtime) as a coder.