Running and cycling to work is a sure bet to get some exercise in even if you have kids.
Those who bike / jog to work, how do you work around this?
Seriously, perhaps bike slow enough not to get too sweaty and smelly? Of course this depends on where you're located...
Also biking doesn't even have to work up a sweat if you don't want it to. But at the least you can ride in a shirt dedicated for exercise (quick-dry, ideally Merino wool) and switch when you get to the office.
And there's a difference between active "stink" and just smelling like sweat and exercise. One is worse but neither is particularly pleasant for your office-mates.
"There's no such thing as bad weather, just bad clothing" does have a limit if outside temps hit 35°C (95°F ish) (wet-bulb temp is more relevant but obviously varies by humidity) or higher but there's a lot you can do to mitigate the effects with the right materials.
The easiest way to get exercise with kids is to take them outside and play with them. Go for hikes, play ball, run around. It's not structured exercise like going to the gym, but it's plenty of activity relative to a sedentary lifestyle.
I think that one reason why people are off putted from exercising is that expectation that everything must be super serious training.