Howdy, Ozzie!
I appreciate and agree with your conclusion about the WTS selective vision and hearing, but I think your example is not a good one for the point.
It seems to me the text you cite is informing of Christian men praying in behalf of a sick friend and also taking practical action to help alleviate suffering by application of oil/grease, which was a common medicinal remedy in antiquity.
I have observed as much in rural areas where experienced and trusted Christian men among JWs were called to the scenes of various sick individuals suffering from radiant burn, dangerous ingestion of hot pepper, and fever. In each case the substances applied contained some or a lot of oil and were given to help impede suffering. I'll never forget the terrible experience of a small child who had eaten raw hot pepper. The chemical burning started in and around the mouth and proceeded to include skin body-wide. The child was in terrible straits of suffering and the parents did not know what to do given there were no medical facilities available and the family was desperately poor. The parents did all they knew to do: they sent for two trusted overseers of their congregation. Both men took one look at the child and immediately took lard from the family's food stores, stripped the child naked and greased his entire body. They also prayed to calm the child and parents. Within about 15 minutes the childs suffering diminished because of the application of lard. Within an hour the child's skin color was back to normal and he was sleeping restfully on clean sheets (also provided by the overseers).
A more contemporary version of the same thing is when experienced and trusted Christian men offer practical help to aid in helping a literally sick person. The proverbial grease on the head might be something as common to us as recommending an application of ibuprofen by an experienced person in the face of inexperience. When this is done by trusted and experienced Christian men who also take the time to pray with a family or individual there is tremendous comfort brought to an individual's emotional and, hopefully, physical wellbeing.
Marvin Shilmer