Healing and Teaching in Matthew's Gospel

From a conversation with colleagues while translating the first part of Matthew, chapter 8:

In Matthew's Gospel, Jesus is always teaching. And yet, it is surprising that following the Sermon on the Mount (chs. 5-7), Jesus heals. One would expect the opposite. However, the pattern in Matthew, where Jesus is strongly portrayed as a teacher, is that Jesus teaches the disicples, but he heals the crowds (oxlos). The "insiders," the disciples get instruction, while the "outsiders," the crowds, get healing.

I wonder if there is not a word for the church here.

I think sometimes we have it all backwards as the church. We as insiders want to teach the crowds: behave, shape up, follow the Ten Commandments! And yet we want ourselves to be healed (What can the church do for me?). The thrust of Matthew's Gospel suggests just the opposite: that instruction is for those who have already signed on as disciples (Matthew 5-7) and the compassion, the healing, is for the world into which these disciples are called and sent (Matthew 28:11-20). Certainly, Jesus does extend healing/compassion to the disciples and also teaches the world, but the primary pattern for me is instructive (no pun): in the church: teach; outside the church: heal/show compassion.

*Thanks also to Fr. Patrick Henry Reardon, whose devotional reflections on Matthew also prompted this reflection.