Is the church doing all it can do to follow Jesus' teaching of loving one's neighbor as one self?

Whatever you mean by 'the church' the answer would always have to be no. This is because people are human (and 'the church' however ones defines it, consists of people) and don't always help others as they ought. Sometimes people also have needs themselves and so don't reach out to others, since they themselves are struggling.

Could 'the church' do more. Of course.