The concept that you are talking about here is called "ellipses" where a few words, already present once in the sentence, are omitted to keep the sentence concise.
Even though ellipses may be applied here, I am a little hesitant about these two sentences. Are these from official sources. I ask this because if I were to write this sentence, I would retain the omitted words - was and were - in these two sentences. Anyhow, I will still explain how the ellipsis concept applies here and then I will also provide a brief explanation of the concept in general.
Once the usefulness of the motor car was proven and its superiority to the horse buggy (was) demonstrated, much of the initial hostility to the motor car disappeared.
The sentence that you have cited as example shows that "was" appeared once in the sentence before "proven" and then was omitted later before "demonstrated" to avoid repetition and keep the sentence concise. The omission of "was" does not affect the meaning of the sentence.
Of all the wild animals in their area, none was more useful to the Delaware tribes than the Virginia white-tailed deer: It was a source of meat, and its hide was used for clothing, its antlers and bones (were) used for tools, and its sinews and gut (were) used for bindings and glue.
In this example too, we notice the same thing. Verb "were" appears once in the sentence and then is omitted later from places where it could be used and this omission does not affect the meaning of the sentence.
Ellipsis in general
We can omit word/words if all of the below are applicable.
a) The word that is omitted later must appear once in the sentence at least once in its entire form.
b) Ellipses is used when there is a uniform list.
c) The omission must not distort the meaning of the sentence or must cause any ambiguity.
Let's see a few more examples:
1: She like to sing, (to) dance and (to) act. Correct.
2: I explored Canada better in this visit than in the last (visit). Correct.
3: Eric is more scared of scorpions than Tom. ???
There is an ambiguity in the meaning. This sentence can have two meanings:
A: Eric is more scared of scorpions than he is of Tom.
B: Eric is more scared of scorpions than Tom is.
In the first sentence, the comparison is between scorpion and Tom.
In the second, the comparison is between Eric and Tom.
Hence, to make the original sentence state a comparison between scorpions and Tom, we must write it as:
Eric is more scared of scorpions than (he is) of Tom.
To state the comparison between Eric and Tom, the sentence must be written as:
Eric is more scared of scorpions than Tom is.
OG12 ques# 87 is one of the examples of ellipsis.
Th concept of "ellipsis" is a very important part of the parallelism. In e-gmat, you may find it extensively covered in our live session on Parallelism. You can also view the recording of the same.
Hope this helps.
(PS: I will wait to hear the sources of the two sentences above)