I had no idea about using it for teaching reading! My work currently involves supervising early-years reading and this could be a great resource. I will look into it. Have you seen it in action?
Yes, I have seen the Psalter and hymn books used in action.
I think my younger sister learned to read primarily from following my finger in hymnbooks as the hymns were sung, and memorizing the scripture printed out on cards for her to memorize each week.
I spent a few years tutoring homeless women and learned a lot about phonics. More than one woman would only read the KJV Bible in their most paranoid states, and I learned how to use that single book to teach a variety of subjects, especially remedial phonics. I could sometimes sneak in a metrical version of a Psalm to pull out a repeated sound or spelling.
The following books will teach you to use poetry/hymns and the Bible to teach reading.
Ruth Beechick's the 3 R's. Mott Media carries other of her books and phonics flashcards that you might find helpful. The methods used and taught in 3R's and the McGuffy readers can be applied to the Psalter and any book.
https://www.amazon.com/Three-Rs-Ruth-Beechick/dp/0880620749
Sam Blumenfeld's How to Tutor. He has a confusing assortment of other books, in different editions with some false advertising. I usually just stick with How to Tutor, but Phonics for Success is my choice to track down and purchase for any student that needs their own textbook and is likely to finish it. The advanced lessons are superior, and the book is more politically correct.
https://www.amazon.com/gp/product/B01KD60MEE/ref=dbs_a_def_rwt_hsch_vapi_tkin_p1_i2
https://www.amazon.com/gp/product/1495144216/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
Ella Frances Lynch teaches children to "read" a memorized portion of Hiawatha. This technique can be applied to memorized Psalms or any text. Start reading "Educating the Child at Home" on page 73.
Educating the Child at Home
Google "Copywork", and look for articles like this one.
The Unlikely Homeschool: Beginner's Guide to Copywork
Don Potter has lots of free handwriting resources. This one is an expanded version of the instructions on the above book, How to Tutor".
http://donpotter.net/pdf/af_cursive.pdf
To use copywork instead of handwriting worksheets, you must use a hand that is made up of repeating strokes, and teach/learn the letters as combinations of strokes, not as individual drawings. Master the hand yourself first, and then you will be free of needing worksheets.
For learning disabled students, I first teach the Spalding hand in manuscript not cursive. Don Potter has a very similar free method.
http://donpotter.net/pdf/shortcut-to-manuscript.pdf. For the more capable students, I teach them to join the lowercase manuscript into cursive.
https://i.pinimg.com/originals/ab/b8/15/abb81507ef8a818ed0eba1cba3757436.jpg but I retain the manuscript for the uppercase letters. I do not teach uppercase cursive. And I do not force remedial students to join their letters. Some disabilites require a student to finish one letter entirely before thinking about the next letter. Curive requires multitasking that is a burden on some students. Lefties, especially disabled lefties do better with vertical instead of slanted hands. The spalding joined letters are vertical. Traditional cursive is slanted to the right.