No. Natural gender means that if a noun’s meaning refers to a male ‘something’, then the noun is masculine, and if it refers to a female ‘something’, then the noun is feminine.
However, not all third-declension nouns have natural gender, any more than other Latin nouns do. Sol ‘sun’ is masculine and feles ‘cat’ is feminine, regardless of the fact that the sun is not a man and a cat can be male. So I am not sure what your textbook meant.
You are indeed missing the point. What case a noun happens to be or, if it happens to be in the genitive, the gender of the noun it qualifies, has nothing to do with its own gender, which (except in the case of nouns of common gender like bos) is a fixed property of that individual noun.
If Cat is a feminine noun then it is feminine
If Sun is masculine then it is masculine
It has nothing to do with ANOTHER UNRELATED WORD (no other
word effects the noun except the fact that it has that gender)
Cat does not effect the gender of ‘a man’ who has something to
do with it.
Natural gender is a vague attempt to make something masc/fem
because it makes sense that way - but no other word has anything
to do with the label masc/fem no matter what the sentence structure
is.
natural gender simply means if you saw it naked what kind of reproductive organs would it have? so an inanimate object would be, by natural gender, neuter.