I imagine that due to scarcity of wood in Ancient Egypt and the very limited availability of such "high-end" chairs, you didn't have to worry about the overwhelming majority of Ancient Egyptian children.
I think you're probably joking, but although we don't know where this stool comes from exactly (at least it's not stated in the british museum's website), it's far more common for furniture that used to ornate palaces and sacred places to survive rather than common household items. So, I think it's likely that no children were playing around such objects.
It would seem to me that common household items could have been even more - much more - dangerous than this. (Also, children used to spend most of their time outside - a significantly more dangerous environment by any measure.)
That could be the case, but I've never heard about any studies about child safety in ancient Egypt. Though I'd like to read that. I really enjoy learning more about how the common folk lived in ancient times, what they thought about, what kind of conversations they had. Unfortunately, we do not have that much to go off of :(