You're absolutely right that the principal motivation for taking in new members is to stabilize the region.
It's the process that matters, to my mind, as much as bringing them in. Before they can join, they have to meet a number of standards, and they work very hard to meet them. They are standards that complement well what the EU insists upon. What we insist upon is: no problems with your neighbours; proper democratic control of the military; transparent systems; interoperable forces, so that they can actually make a contribution when they get in; etc.
We push very hard for them to be like us in terms of their own internal systems and their relations with their neighbours before they get to walk through the door.