I know I’m late to the thread but here goes…
First, yes, the Sacrament is “reserved for people with serious illnesses.” There’s no question about that. It’s not for minor illnesses, and it’s not for the healthy. Defining “serious illness” has been a question ever since Pope Paul VI promulgated the new rite of Unction/Anointing in 1972.
There is a special Mass text to be used when Anointing is done within Mass. The prayers (collect, offertory, post-Communion and preface) are all unique to that Mass, and there are special insertions into the Eucharistic Prayers, and a special introduction which is done before the Penitential Rite. If you have the book “Pastoral Care of the Sick” it’s all described here.
The actual Rite of Anointing is done after the homily (or Gospel).
Anyone may participate in this Mass, but ONLY the seriously ill may be anointed.
Here’s a good resource from the CDF:
adoremus.org/CDF-Healing.html