As someone who grew up i a Lutheran country I can tell you that to me this does that Bible quote not at all look like anointing of the sick, the so called “last rites”. I realise that is not the name you have for it, probably because you find it misleadning, but still. That quote from the Bible to me looks like what people in charismatic churches would use a lot - the Pentecostales, etc. Those who claim that the Holy Spirits makes wonder today, in healing people. In Catholic faith people get healed from praying, asking saints to pray for them, right? Of the Lutheran churches some don’t stress it much at all, and don’t think it happens much, but when healing is performed in (I hate to call them Lutheran. What word should I choose… “post-reformation”?) churches, it is typically done by a pastor or someone supposed to have been given the gift of healing laying their hands on the sick wanting to be healed, usually accompanied with speaking in tongues and other extatic expressions. Personally I am very sceptical of this particular church variety, but I am sure these Christian would question you on these Bible verses: do the sick who get anointed actually get healed?
Reconsider your reading of the passage. With the ritual there are enjoined three promises: (1) the salvation of the sick, (2) his raising up by God, and (3) the forgiveness of his sins.
Salvation can mean the healing of the body, but it more primarily means the salvation of the soul. Remember the Lord’s counsel to not fear those who could destroy the body but rather those who can destroy both the body and the soul?
Raising of the person could mean the raising of his body in the sense that he he is healed well enough in order to rise out of bed. I would argue that it more likely refers to the final resurrection, in which all will rise to be judged, but the just to eternal reward and the wicked to eternal punishment. For that reason, the righteous can be said to rise again in a more excellent way. The wicked, although they are restored in body, can be said not to rise, but to fall to their eternal destruction. Consider the words of the first Psalm where for that reason it attributes rising to the just only:
The unjust will not rise in the judgment, nor the sinners in the council of the just, for the Lord knows the way of the just but the path of sinners will perish.
Finally, the forgiveness of sins can occur without the healing of the body.
That is not to say that physical healing cannot occur with the sacrament. This is from the 1582 Rheims NT Commentary:
Shall raise him up. When it shall be good for the salvation of the party, or agreeable to God’s honor, this Sacrament restores also a man to bodily health again, as experience often teacheth us. Which yet is not done by way of miracle, to make the party suddenly whole, but by God’s ordinary providence and use of second Causes, which otherwise would not have had that effect, but for the said Sacrament. This is the second effect.
sites.google.com/site/aquinasstudybible/home/james/douay-rheims-study-bible-on-james/chapter-1/chapter-2/chapter-3/chapter-4/chapter-5
Although the sacrament of extreme unction can cause the healing of the sick, it would not be fitting if that occurred in every case regardless of circumstances for the same reason that baptism ought not to prevent the death of the body. If carnally minded people who otherwise despised holy things saw the miraculous effects of the sacraments, they would seek them out for carnal reasons and not for holy reasons. And besides that, we all have our time to die eventually. St. Paul says in the Second Letter to the Corinthians that he would rather be with the Lord in heaven than on earth in his body. It wouldn’t make sense if Christ instituted a sacrament that always kept someone from dying and commanded it to be used in danger of death whenever possible. Sacraments are supposed to help us get to heaven, not stop us from getting to heaven.