I can understand your situation very well, as I have a similar situation. I carry my 3 girls (age 6, 2.5 and 2.5) and want to use the car less. Some of the hills in my area I can barely make it up with everyone, and while we can do most of them, it's a slow process to get anywhere involving a lot of granny gear. I have a moderate level of fitness - not as good as I used to be when I had time to mountain bike all the time, but not too bad! It was a real delimma for me, because I've always been a big bike commuter and don't like driving, but I'd never had to carry so much weight. I just found it really hard to get motivated, especially if I was feeling tired (not unusual with 3 small children) or running late (often). We live about 7 miles from the center of town, so it's a good 15 mile round trip just to go to town. The car was so easy....
So I decided to get the electric assist on my bike and it's made a huge difference! I know there are lots of people out there who do it without, and those people rock! But for me, I just ride the bike so much more with the assist than I would without. The practicalities of life just took over the idealistic nature that I didn't need it. And I'm still saving huge amounts of energy by not driving the car. (the most annoying people are those ones who think the assist is cheating somehow but then drive their cars 3 miles to the start of a road ride - I don't think I'll offend anyone on this list by saying that but if I did, too bad!).
Anyway, now taking the bike really is convenient and that makes it competitive with the car. I still get a little workout (I choose how much) and it takes a little longer than driving (a LOT faster than without the assist), but the benefits of having happy kids in the back rather than them fighting in the car is incredible. What is it with the car? On the days I do drive, I often arrive home frazzled and annoyed, while that is not nearly as common on the bike, where they seem less prone to needle-ing each other. I've become sort of an evangelist for electric cargo bikes because it's really touched my life in such a big way. I used to drive every day. Now I drive about once a week. When I started looking into the energy used in driving, I became even more of an evangelist because cars are just such energy hogs. It motivated me even more to keep doing it!
I know getting set up with a cargo bike, and especially adding electric assist is a pretty big up-front cost. But if you look at it as replacing a car or car trips, I think it makes it seem less so. For me (and I suspect many others here), the benefits of biking vs. driving are so huge, that it becomes worth it over time. I wouldn't go back now that I have the setup that I do. In fact, I'd sell my cars first. People see me riding with my bucket full of kids and think I'm a super-woman. Sometimes I tell them about the assist - sometimes I just let them think that : ) But mostly, we just enjoy the riding. Cheers to all the super women (and men) who are out there now exposing our kids to a great lifestyle. I hope you'll decide to join us one way or the other!
You can check out my video blog about lifestyle biking on my website - here http://www.cycle9.com/category/go-by-bike/