Izkata's answer is the most important here: there are many references to Human-led events and cultural works between now and the period in which Star Trek is set. But I'd also like to add that, from the perspective of the characters on the show, the distinction might not be as important as it seems to us. Remember that Humanity is now just one race among many in the Federation: there may not be as strong a distinction between "Human" culture and Federation culture at large.
Imagine a citizen of Richmond from the 1700s looking at modern American culture and exclaiming, "what happened to the culture of Virginia? No one ever talks about the achievements of Virginians, or of the great artistic works coming from Virginia! Do Virginians not have any culture at all anymore?"
It's not that such a person would be right or wrong, it's that modern American culture is simply more homogenized than the question assumes: an author or musician from Virginia today might move to New York or California after college without a second thought, and will generally just be referred to as "American." Specific genres or styles may be associated with certain regions, but generally artists of the 21st century aren't strongly associated with their home states. (Hell, many Canadian actors and performers go their whole lives with their audiences not even realizing what country they come from!)
Similarly, "Human" culture in the 24th century might not be terribly distinguishable from Federation culture in general, and the crew of the Enterprise may not see any reason to make that distinction. Undoubtedly there are still great Human artists, but just as a list of Country music singers today might casually omit any Texans, a list of great Federation artists might casually omit any Humans.
