The very idea of helping travelers in trouble was based on a concept of Buddhist thought called 'kyujutsu' (helping a person in trouble), so temples naturally made up the majority of the operations.
元々困っている旅行者を施設を造って助けようという「救恤」の発想自体が仏教の考え方であり、自然と運営主体は寺院が多くなった。