One of the benefits of shooting portraits in people's own homes is the variety of locations and backgrounds that can be used to give a range of styles and moods from one single shoot. By taking your children to a photographer's studio you are immediately limited by what that studio has to offer.
For me the best portraits offer some context and show the child in their own environment. This also has the benefit of putting them at their ease in front of the lens as opposed to the artificial environment of a studio with all it's umbrellas, flash heads and backdrops.
These two images are a good illustration of this showing how much variety you can get from having a photographer to your own home. The close up portrait is shot in the child's garden with natural light and has a similarity to a shot you might expect to see from a studio shoot. But then a short walk from the house was a small loch where the children loved to go exploring and looking for tadpoles and pond skaters - no studio shoot can ever hope to offer such diversity.