Well it is a totally strange thing to do. I can't understand how removing "auxiliaries" helps you to understand English. Removing the auxiliary verbs completely changes the meaning of the sentences. But....
There are two "simple" tenses in English "simple present" and "simple past". So the question seems to be "find the verb phrase, and replace it with the simple present or past" Other tenses, moods and negations are formed with various types of auxiliary verbs.
The verb phrase is emphasised.
I have wanted to see the Grand Canyon.
I see the Grand canyon.
The second one is rather odd. Since in English, negation is done using an auxiliary. However if we replace the verb phrase with the simple form of the verb we get:
My nephew hadn't repaired his car yet.
My nephew repaired his car yet. (incorrect sentence)
But now we have a problem, since the adverb "yet" isn't used with the simple past like this. We can't answer this question unless we either change "yet" to "already" or use an auxiliary to form the negative. The book answer keeps the auxillary "didn't". If we "rewrite appropriately" we get
My nephew repaired his car already.
(this particular sentence would be better in the present perfect, that that tense is formed with an auxiliary)
There is a negation that doesn't use an auxiliary, but it is archaic: "My nephew repairs not his car yet". Obviously this is not good modern English.
Since auxillaries are used so much in English, and since some adverbs can't be used with some verb phrases, this kind of problem will always occur. There is no way to "remove auxiliaries" from the second sentence, and without changing other parts to fit.
So "Why (in the answer key) is tense of the original verb not the same as verbs when we had auxiliaries?" Because removing auxiliaries changes the tense. Indicating tense is one of the many functions of auxiliary verbs. In the answer key, sentence 1 uses the simple present, not the present participle. Sentence 2 uses the negation of the simple past, not the past participle.
The additional examples that you have now posted make no sense.
The book is completely contradictory and this exercise is entirely useless.