Most screens are very lightweight. This makes it easy for them to be moved when required so that you can load wood into your stove. It also means that if they fall over on a child they will not be harmed. The sound of a screen falling over will scare the child and alert the parent to check on things.
Screens are also important for adults as well. It is far too easy to brush up against a red hot stove and get a burn. This is especially true when you have company visiting who may not be used to a woodstove.
A proper screen may also assist in preventing flammable materials from coming in close proximity to your woodstove. Once again children have a tendency to throw balls and dolls and all manner of things in all directions. A screen will prevent these objects from coming in contact with your stove.
A screen does not need to be large and intrusive. Many are very decorative and ornate. The consequences of not having a screen are severe enough that nobody should consider burning a woodstove without one.