Let me guess, your spouse isn't in a medical residency?
Before I had kids, I never had trouble finding time to exercise and had little sympathy for people who did. Since I've had kids my ability to exercise has varied with life circumstances. At the moment it is very hard for me because my wife is in a medical residency, so most of the time I'm an only parent to small kids.
In particular in periods when I exercised, I've found that between my personal rhythms and outside climate, morning is the only time that it makes sense for me to do so. (Perhaps you are happy running in 100+ degree heat, but I am not.) But when I'm on my own trying to get 2 kids up, ready, and out the door, before I start doing other stuff, I don't have 20 min where I can safely disappear on them.
Not many people's spouses are in medical residency, and have children who are too young to be left unsupervised for 20 minutes.
Even if you broaden it a bit (single parents, spouse in the military on deployment, etc) you are not looking at 90+% of the population. And most people looking after small children are pretty active anyway (due to all the lifting, running around, and so on).
Also in every family with small kids, if one parent wants a piece of freedom, it puts pressure on the other parent. So even if it is theoretically possible, if half of those parents get the freedom to exercise in the morning, the other half do not.
Does it amount to 10% with a legitimate difficulty in scheduling exercise? I'm not sure. But it is more than I would have guessed before I had kids.
Staying as a single adult in the house is a long-term choice.
You can: marry, invite your friend to live together, invite your parents, rent our a room or two in your house and ask for occasional babysitting help from them.
1. I can do HN with the kids in the house and use my ears to supervise. I can't do that with exercise. (My son is right now sitting next to me, counting and recounting his allowance.)
2. I can comfortably do HN while it is too hot outside to exercise.
hm. a machine solves both problems; I mean, I don't have kids but most of the cardio I've done in the last 5 years or so has been on a machine; instead of watching children, I watch silly action movies. For me? it's to distract me from the monotony. Action movies were made for my attention span; but it seems you could just as easily supervise children.
That said, I certainly feel a lot more difference from resistance training than from cardio, and diet changes have been more effective than either in controlling my weight. (e.g. an hour a day of running is not going to keep me from ballooning out if I use mountain dew as my primary caffeine delivery mechanism.)
I mean, you can get a really nice treadmill for a grand and a half; by the standards of medical care, that's really cheap; Personally, I've also had some success using the manual type (which can be had for $50 or so on craigslist) unlike most other cheap cardio equipment (I utterly destroyed a rowing machine through a month of daily use... most low-end cardio equipment wasn't designed for actual use by a 200lb man.) the manual treadmills seem to be fairly durable. "I'm going to go push the shopping cart" I'd say. It does take a bit more concentration than an automatic, just 'cause if you stop, it stops, but eh, it's certainly usable, and you get a little bit of upper-body work done, too; it really is very much like pushing a heavy shopping cart.
Calisthenics can work well while watching kids. With no setup or teardown time, they work nicely for opportunistic, "hmm, looks like he's going to be occupied for a few minutes" holes in the day.
We also went running with a jogging stroller when the weather was nice. Our son usually loved the ride, and was asleep by the time we got home (bonus!). That might not work on its own with kids, but one of the places we went running was a local school with a track and a playground. I know the kids in this house consider a 30 minute visit to a playground far too short, and that's more jogging than I can handle.
My daughter needs over 12 hours of sleep, and has trouble falling asleep before it gets dark. I'm not going to make her get up earlier without very good cause.
Also I would not trust you to supervise preschool children in a public space. You clearly do not know what is required to do so responsibly.
Over-supervising children hurt them more than helps. Children need their freedom too.
Besides, if you do not exercise - it's bad for your children. It's especially bad if they do not exercise either (because you don't have time to play with them, and don't allow them play alone, because you want them to be extra safe).
They are all getting ready for school in the morning themselves. And besides, have you ever tried convincing teenagers to get up earlier in the morning? (The ones that I know around here would prefer to sleep in.)
My now teenager wakes up for school herself at ~6am starting from ~age 8.
Though I know that other kids might be different. But you need to find only one babysitter.
How many teenagers in your neighborhood did you ask?
Besides, there are not only teenagers, but housewives who wake up in the morning anyway.
Or how about this: you exercise while your wife and your daughter are sleeping (or your wife is getting ready to go to work. Or while your daughter is sleeping alone.
Or take your daughter on a run with you in a stroller. She may keep sleeping if it's an issue to wake up.
You may ask your older kid to babysit younger daughter while you are taking 20-minutes run with your cell phone and give another cell phone to your older kid.
You may take two 10-minutes runs if you worry that kids cannot be left alone for 20-minutes.
There are so many ways to find a way/time to exercise ...
Why would kids stop you from getting exercise?
Get a stroller and give your kids a ride while you are running.
Or give them bicycles and run with them.
Not sure why we all should follow "1 hour runs 5 times per week" suggestion. We are not professional runners.
And note that in this thread the standard for what you should do is an hour a day 5 days a week rather than 20 minutes a day for 7 days a week.