I have never heard of this method, so I can't tell if it really works. I also have no idea how to interpret those explanations. If there is something missing in the book, i normally would recommend to ask the author, but since Mrs. Pollard war born 1902 (her book is from 1957), this might be hard.
I just can recommend to analyze the grammar of the German sentence, to find out what belongs together.
This sentence is neither a question nor a command, so it is a statement, and in a German statement there is one very hard rule:
The finite (i.e. inflected) part of the verb always stands on position 2 of the sentence.
All other parts of the verb (if there are more parts) are located at the end of the sentence.
So, everything before the verb is one single part of speech (well, there can be exceptions, but they are rare). So these 12 words (everything before the first verb!) are one part of speech:
Die wesentliche Verschiedenheit der dauerhaften Güter einerseits und der Nutzungen derselben anderseits
And this sequence contains a noun in nominative case (die Verschiedenheit), and since there is no other word in nominative case in the whole sentence, this word must belong to the subject. So the 12 words at the beginning together form the subject. Let's think of it later in detail, and for now let's replace it with a personal pronoun, that matches with the core of the subject (die Verschiedenheit) in number and gender:
Sie tritt in diesen Fällen langer Dauerhaftigkeit besonders deutlich und klar hervor.
So, now he have a subject (sie), and the finite part of the verb (tritt, infinite form: treten). Let's see, if the last word is something, that belongs to the verb. We find hervor which can be joined to treten, giving the separable verb hervortreten.
Sie tritt ... hervor.
This is Präsens (in most cases equivalent to present tense). To test, if the two words really are parts of a separable word, we can convert this sentence into Futur I (equivalent to future tense). Then the verb will appear in its infinite, i.e. unseparated form, because now the auxiliary verb werden will be inflected and has to occupy position 2:
Sie wird ... hervortreten.
So now we have the complete subject (12 words) and the complete verb (2 words) the rest must be predicatives or objects.
In German grammar we only call tritt together with hervor the Prädikat. In English Grammar, the term predicate has a wider meaning. Also the parts of speech that I called "the rest" would belong to it, but in German grammar they do not.
Prädikat
We found out, the the Prädikat is
hervortreten
dict.leo.org offers some possible translations:
to emerge, to abound, to step/come forward, to bunch out
Subjekt
Die wesentliche Verschiedenheit der dauerhaften Güter einerseits und der Nutzungen derselben anderseits
You already translated it correctly. I offer just another variation (I also would have used zwischen + Dativ instead of a genitive construction in the German sentence):
The essential difference between the durable good on the one hand and the utility of the same on the other hand
"the rest"
The rest is a Präpositionalobjekt (prepositional object)
in diesen Fällen langer Dauerhaftigkeit
and a Prädikativum (predicative expression)
besonders deutlich und klar
The prepositional object translates to
in these cases of long durability
and the predicative expression contains the fixed phrase »deutlich und klar« which you find more often in reverse order (»klar und deutlich«). But the two adjectives are synonyms, they mean the same. So literally it would translate as »clearly and clearly«, but that doesn't make sense in an English sentence. One »clearly« is enough. So »besonders deutlich und klar« just is
very clearly
complete translation
So let's join together what we have:
The essential difference between the durable good on the one hand and the utility of the same on the other hand emerges very clearly in these cases of long durability.