Who is at fault for Elesin’s failure to fulfill his duty?

Pilkings is at fault for Elesin's incapability to fulfill his duty. 

Elesin is at fault for not fulfilling his own death. He was too attached to Earthly pleasures (lust, women). Even if Pilkings had not sent his men, he would not have killed himself. This is foreshadowed when the Praise- Singer tells him about the story of the Not-I bird. This story talks about how some turn away from the call of death.

Hope it helps!