Why do we remove the article “the” when adapting sentences that express belonging when we write them using the genitive’s case?
Example: Many poets were inspired by the beauty of nature.
Many poets were inspired by nature’s beauty.
Why do we remove the article “the” in this kind of situation?
Thanks in advance!
The definite article helps identify which, Here in your example, the underlying question might be “which beauty?”
In “nature’s beauty,” the possessive of “nature” helps answer that. The use of “the” would be considered informationally redundant.
It’s a simplification that’s developed independently across many Germanic languages over time.
There is no definite article in front of the word nature in your first sentence, too. So instead of thinking of it as removal you could also consider it a replacement of the the with 's. 
As a native English speaker, I have no idea except to say that they sound correct as they are. Sometimes it is best to just learn the patterns. I realise this doesn’t answer your question.