"On The Other Claw...."

Yeah... well, it is rather more likely that people in the Trek universe would be speaking something similar to 20th century English, because the basic words of the language don't change as fast as they once did. I think that's because of literacy. Written language tends to freeze in place somewhat.

Consider that Chaucer died in 1400. Shakespeare was writing his plays around 1600. It's been 400 years since Shakespeare, yet we can still pretty much understand his English. Chaucer's, on the other hand, is a foreign language even though it's only half again as old. That speaks to the language changing less in recent years than it once did.

But the reason the People of the Future speak 20th Century language is the same reason Enterprise has those magical "universal translators," not to mention transporters, warp drive, faster-than-light radio, and all the rest of it. It's all because of plot necessity. Yeah, they could have spent 20 minutes of each show landing an 1100-foot starship on the planet's surface, spent thirty or forty years of ship's time travelling from one star to the next, or had everyone aboard jabbering in some 24th Century Cityspeek of polyglot English- Spanish- Chinese- Japanese, a' la Blade Runner. But if they did stuff like that they'd have to spend the whole episode getting everything set up and furnishing translations for the viewers, and the hour would be up just when Kirk set foot on the planet to go meet the Alien Babe of the Week. Wouldn't make for good ratings.