Her responsibility is that these things happen according to expectations and decisions of the other deities. Demeter had only one child Persephone. Probably because it means death-bringing and it relates to her position as Queen of the underworld. This is true if you understand living in the underworld means dying and winter is the cold time of the year when plants die back. She could also move into Hades and out again, she could determine the fate of souls, She could bring things back to life.
Goddesses have no enemies.
Also it is not clear that Persephone wanted to escape Hades. Though Hades used deception to keep Persephone, it may only have been Demeter who was deceived. Persephone and Hades are a devoted couple. Demeter becomes unhappy with the result the a cold, dry season comes to the countryside.