Why is rap the only genre of music that actually cares about who the best is?

It's not. The strive for excellence exists in all genres of music. It's just that rap seems to be severely lacking in the musical ability and songwriting categories. So, I can see how the assumption can be made, when you are used to hearing a type of so called "music" that has none of these elements.

Talking. Yes, it's a skill. But it's not art.

Your premise is incorrect.

Although I believe that artists shouldn't compete, the Grammy Awards have been recognizing the "best" in several musical styles for decades.

You'll have to explain this a bit better, not sure I understand.

Debussy is my savior