Late reply, but we got rid of DirecTV over a year ago. We use Hulu and Netflix exclusively, no cable or satellite. We're very happy. However, neither of us watch sports. I watch the NBC nightly news through a podcast.
The initial catalyst was the price of DirecTV. Our friends got some great deal as new subscribers, so my husband called and asked for that deal. They said No. He cancelled. A month later, we got an offer for that same deal. By that time, we were not missing DirecTV at all, so declined.
So go ahead and cancel. If you are miserable, just wait for the offer to return. We're still getting the "Come back!" offers, nearly two years later.
Oh, to answer your question about kids programming. Netflix may not have the programs your kids are used to, but they have plenty of shows. With the new Kids Only area, they can tool around and find something new they are interested in.