The most obvious source of additional help would be deacons. The obstacle, however, is that the Council of Trent (based on James 5:15) said the sacrament of the sick forgives sins—and of course we can’t have deacons doing that! But how many of the sick are even aware that this sacrament is a twofer? The idea made sense when anointing was called “extreme unction” and only administered at the point of death. It makes little sense today. The Second Vatican Council rightly restored anointing of the sick to the status of a repeatable sacrament for those who are seriously ill, impaired by old age or chronic illness, or facing surgery. For those at the point of death, the path is clear: they are anointed, their sins are forgiven, and they receive Communion (viaticum)—a trio the Catechism likens to the three sacraments of initiation. It is their preparation for “passing over” from this life to the next, just as initiation is a “passing over” to new life in Christ. But for others who are sick—and this constitutes the majority—a different approach makes better sense. Oil for healing. Laying on of hands. Prayer, faith-filled encouragement, compassion.
The proposal makes little sense and the argument for it makes none. Nothing in this paragraph shows any understanding of how sacraments work. Sacraments are signs, so how they are done is important; and, in particular, the ministers of the sacrament and the circumstances in which they can be ministers is part of the sacramental sign itself. If you don't have the right minister(s), you don't have the sacrament at all, just as you don't have the sacrament if you don't have the right matter or the right form or a context showing that it has the right end. The minister of the sacrament is given in James 5:14; there is nothing whatsoever that indicates that anyone else can do it, and the Church statements insisting that only the priest can do it are quite easy to find. And this, indeed, makes some sense in terms of the other features of the sacramental sign itself. The Anointing of the Sick signifies the resurrection of the dead and life in the world to come; any healing in it is a small foretaste of glory, both in sign and in grace, spilling over from Christ at the right hand of God. That it is only given by the priest, whose entire sacramental existence is concerned with participation in the heavenly liturgy of Christ our High Priest in heaven, contributes to its signification as the sacrament of glory.
But that is a relatively minor matter; people can easily be confused about who can be minister. What is egregious here is the "twofer" comment, which shows a completely failure to understand the structure of the sacrament of Anointing of the Sick itself. Forgiveness of sins is not a 'twofer', as if it were some additional part; it is integrally connected with the healing. In the sacrament, grace overflows from Christ to the soul, thus fortifying or reviving its health, and the consolation of the soul also overflows and fortifies or revives the health of the body to the extent possible. That's the whole point. Healing of the soul is the first and immediate effect of the sacrament; healing of the body is something that follows on this insofar as conditions allow. And healing of the soul includes forgiveness of sins. It's one and the same grace that does both; there is no 'twofer' here. You can't have one without the other and still have the sacrament. And even setting that aside, what people are 'aware of' does not affect the nature of a sacrament -- that is one of the most fundamental and basic principles of Catholic sacramental theology.
Likewise, the Vatican II argument makes no sense whatsoever. It is very possible to exaggerate the extent to which the understanding of the sacrament has changed -- contrary to what Ferrone's words seem to suggest, it was always regarded as a repeatable sacrament, for instance, and the sacrament still serves in the same 'on the verge of death' capacity now that it did; if you receive it instead because of old age or chronic illness, you are receiving exactly the same sacrament that you would get in a case on the verge of death. That was the whole point of insisting on the point in the first place: that the very same sacrament could be applied to more than just nearly dead. It's just extraordinarily strange: Ferrone wants something that has a different ministerial requirement and works in an entirely different way from the Anointing of the Sick, and yet she still insists on talking about it as if it would be the Anointing of the Sick.
There are signs throughout that Ferrone thinks using oil for prayer in cases where people are sick just is automatically the sacrament of Anointing. For example, she says,
We already allow deacons and lay ministers to bring Communion to the sick, and the Eucharist is the premier sacrament, to which all other sacraments are ordered. Why be so stingy with oil for the sick?
But deacons and lay ministers aren't the ones who actually do the sacrament of Eucharist; they are just distributing. They can do so because when the priest ministers in the sacrament the sacrament is then by its very nature distributable. But the oil for the sick is not the sacrament of Anointing. It's merely the matter for it. The sacrament only happens when the oil is actually applied with the appropriate kind of prayer by the priest. The parallel would not be laity and deacons distributing Communion but laity and deacons confecting it -- as if the deacons and lay ministers carried around blessed bread and wine and then actually tried to celebrate Mass themselves when they got to the sick. This would not be the sacrament of Eucharist. It is entirely possible to have anointings that are not the sacrament; people do such things with blessed oils all the time. They just don't pretend that it is the sacrament.
[Incidentally, Ferrone gets the Council of Trent wrong on the subject. Council of Trent does not merely say that the sacrament has to do with forgiveness of sins, therefore only priests can do it -- after all, baptism also has to do with forgiveness of sins, and in principle anyone can do it. The Council of Trent specifically says that Scripture only indicates that priests can do it (James 5:14, not James 5:15) and that the Church carefully does not do anything other than what St. James says on the subject. But you could indeed make an argument from forgiveness of sins here. The Council explicitly says that the sacrament is the completion of the sacrament of penance, so as such the sacrament should not be less restrictive in its possible ministers than the sacrament of penance. (It should require ministers at least as consecrated.) But the sacrament of penance or reconciliation is reserved to priests.]