but Star Trek made a certain kind of sense in the late 1960's (nearly 50 years ago!) when the U.S. and the Soviet Union had real space programs which tried to do new things, one after another.

I haven really watched more than a few episodes of ToS, but IIUC it never even bothered to be a realistic depiction of how space exploration would look like. It was more e metaphor of the Cold War, in Space!

would people 50 years from now, in a permanently Earth-bound reality, bother to watch these ancient shows and obsess over the characters?

They will probably idolize some dude who played a vampire. Or zombie. Or BDSM vampire zombie...