Consider this exegesis:
I don't think the passage is about physical healing for several reasons:
1) There's no where else in Scripture where Paul called for elders to exercise the gift of healing. This presupposes healing as a necessity for eldership. Elsewhere, Paul told Timothy to drink wine, not call on elders. Elsewhere, Paul himself prayed for relief, not calling on elders to anoint him with oil.
2) The whole of James is talking about patience and perseverance in suffering. Wuold James suddenly give counsel to avoid suffering and get healed quickly? It doesn't fit.
3) Let's look at three words and their usage in the NT:
A) SICK - used 34 times in NT (20 in gospels/Acts and usually refers to health; 14 in epistles and usually refers to weakness, discouragement, exhaustion, spiritual health, etc).
See 2 Cor 13:3 - He is not weak in dealing with you, but is powerful among you
See 2 Cor 12:10 - For the sake of Christ, then, I am content with weaknesses, insults, hardships, persecutions, and calamities. For when I am weak, then I am strong.
See I Cor 8:11 - And so by your knowledge this weak person is destroyed, the brother for whom Christ died.
See Romans 14:1 - As for the one who is weak in faith, welcome him, but not to quarrel over opinions
So, from this word usage, we see that it doesn't necessarily ALWAYS mean physical illness. THe context of this passage would suggest that the alternate meaning is to be found here. James talks of sins and forgiveness and confession.
B) SICK - a different word for "sick" is used in v15 - KAMNO. This word is only used in Heb 12, where it says "Consider him who endured from sinners such hostility against himself, so that you may not grow weary or fainthearted." It is the word "weary."
C) HEAL - the word is AIOMAI, which is used in Hebrews 12:3-13 at the end of a long passage on spiritual discipline. Not healing from physical illness but refreshment.
D) ANOINT - this is a fairly vulgar use of the word which means "to rub". Used as in case of woman anointing Jesus' feet with oil in Luke 7. Not so much for medicinal purposes but for refreshing purposes. Again in Matt 6:17, Jesus says to anoint when fasting so you won't look like your fasting.
4) JAMES' ILLUSTRATION - James chooses an illustration from the life of Elijah to illustrate his point. He chooses one from I King 19 about when Elijah was discouraged and weak after running from an evil king and queen for years and years. If James wanted to illustrate physical healing, he did a poor job. Plus, just a few passages earlier in chapter 17, Elijah DOES heal someone.
All together, this passage does not seem to be about physical healing from physical illness. You can't use this passage as a FORMULA for obligating God to heal you just by using oil and praying. God does not bend that way. Instead, it is about people struggling and stuggling with sins and rough situations and life who have grown weary. They are to come before the leaders of the church and be REFRESHED, healed in their spirit.
I hope that helps.