§ 395. Responsibility of public welfare districts for the welfare of children. A public welfare district shall be responsible for the welfare of children who are in need of public assistance and care, support and protection, residing or found in its territory, insofar as not inconsistent with the jurisdiction of a family court. Such assistance and care shall be administered either directly by the public welfare official charged therewith, or by another public welfare official acting on his behalf by and pursuant to the provisions of this chapter, or through an authorized agency as defined by this chapter.