1.) When talking about the body of Christ, Paul is first talking about spiritual gifts. Then in I Cor. 12:13 he is clear who he is talking to. He says "For by one Spirit we were all baptized into one body". The setting is, these are believers, spiritually reborn, infilled with the Spirit. Paul said not all are of the faith.
Then in the same passage, speaking of "unseemly" members, he goes on to talk more about the gifts of the Spirit and therein draws this distinction - some members of the body have more "out-front", visible, "desirable" gifts; while others' gifts are of service and support, "in the background", seemingly less important if looked on the surface and carnally. However, Paul says every member's role is vital to the functioning of the whole body, none more important than the other in the church/body of believers.
2.) I believe a separate issue is, as related to the OP, when Jesus spent time with sinners, His Presence was to call them to repentance and therefor heal them (the Great Physician, Mark 2:15-17).