Yep, that is pretty much the process Engblom is using.
Moving back to your question, I am nearly certain the toothpaste recipe is an adaptation of an old one from the mid 1800s or early 1900s. I get the feeling the author might not have done his/her research thoroughly enough and dropped the ball on translating "neutral soap" into modern day terms.
What is meant by "neutral soap" in this context would be a mild, well cured soap that has no flavor of its own. Back in the day, many soaps were slightly lye heavy and often made from rather unsavory types of fat. One paid extra for a well made "neutral" soap that was suitable and pleasant for the toilet and bath. Today, "neutral" soap in this sense is the rule, rather than the exception.
A pure olive oil soap is classic for this purpose, based on what I'm seeing in the old books, but a 100% tallow soap was also used for tooth recipes. Most modern tooth-soap users avoid soap with any amount of coconut oil in it, because CO soap has an objectionable taste. You might already have a suitable "neutral soap" in your stash!
Here's one source with a number of recipes similar to yours: A Practical Treatise on the Manufacture of Perfumery. C. Diete, author. W Brannt, translator. 1892. Chapter 11 covers Dentifrices, mouth-waters, tooth pastes, tooth powders, etc. Source: internetarchive.org
Here are some quotes from this book that seem to relate to your question. I added the comments in brackets [ ].
"...In accordance with recent medical directions and opinions, soap is again employed, and justly so, for the better cleansing of the teeth, whilst formerly it was generally considered injurious. However, though soap is innocuous to the teeth, it should be used in very limited quantities, since its introduction into the mouth is repugnant to many persons, producing in many cases vomiting. The quality of the soap must also be taken into consideration, and only the best neutral soap in the form of a powder, such as is used for fine milled soaps, should be employed...."
"...Odontine paste. —French chalk [aka calcium carbonate] 30 drachms, soap 15, sugar 15, gum-arabic 2, peppermint oil 2, glycerin 8, water 8. Work into a paste...." [All the units in this recipe are drachms. 1 drachm = 4.37 grams]
"...Tooth-soap. —Castile soap 1 lb., prepared chalk 1 oz., thymol 20 grains, oil of wintergreen 30 drops. Shave the soap into ribbons, beat it into a paste with a little water, and add first the prepared chalk and lastly the thymol and wintergreen oil dissolved in a little water...." [1 grain = 0.065 grams]