So you have said, but I know these types of doors from the days I was a security guard going from building to building. These you can't just pull behind you to lock it again. I'm certain of that. I'll go down to the hardware store and check it out.
I agree that a small child could open it, but I dispute your concept of just pulling it to close it behind you.
Have you not looked closely at the two bolts on the door mechanism?
The bottom bolt is called the latch bolt and is tapered in order to allow it to engage the strike plate easily. It is this bolt which engages when the door is closed. From the inside you pull the slide to open it. From the outside you need the key to retract it.
The top bolt is square in profile and is called the deadbolt. This bolt has to be engaged and disengaged with the key.
From this you must surely understand that the key is always needed to enter from the outside. The key retracts both the deadbolt and the latch bolt when turned fully.
Most door locks allow them to be pulled shut, the Chub for example has to be locked back to disable this function. However, the default setting allows you to pull the door closed behind you. People don't want to have to physically lock an outside door every time they leave a property.

