But one thing that I always think is really weird when watching it is that all the cultural references are things that would be familiar to a late 20th century NPR-listening American. One of my favorite book series (Honor Harrington by David Weber) uses a lightly different period. It's references are almost universally to things that would be familiar to people who spend a lot of time with late 18th the early 20th century Western Military History.
It seems weird, but in a lot of ways that's the point. Star Trek isn't a sophisticated imagining of how culture could change if certain technologies appeared. It's about how a polity built on principles every 60s liberal would love (including a fairly muscular, militaristic, foreign policy that a lot of current liberals hate) acts IN SPACE. You don't hear anything about post 20th-century culture, shit that happened outside the main storyline, internal Federation politics (ie: who did Kirk vote for? why?), economic matters (for example once replicator technology exists almost all sectors of the economy are obsolete, because instead of spending months raising a chicken you can spend 2 seconds beaming a perfectly cooked chicken breast into existence, yet half the time they act like the economy is identical to the current US economy and the other half it's a socialist utopia), etc. It is barely Sci-Fi, because (unlike Star Wars) it actually cares how the technology works, and occasionally has story-lines based on said technology (ie: Riker gets cloned by a Transporter, every one of those hateful Holodeck episodes, etc.).
Weber's Honorverse is a bit more Sci-Fi, because he has actually put an awful lot of thought into precisely how the tech affects the culture, but he designed the tech specifically so that he could do things like create a massive ethnic Chinese Empire based on Frederick the Great.