Trek aired on WPIX well into the 90's, but not as often toward the end of their rights period. You had to look for it and hoped for the best. Then the rights were grabbed up by the Sci-Fi Channel which had it until 2004 (they were part of the basic cable package long before this so many people had access). Then TV Land and G4 ran the series. Finally, TOS-R hit the air all over the dial in NY.
So, really, aside from a relatively short gap at the end of the 90's, Trek was always available in the New York City area, it just left our local syndie stations when SFC bought it. It certainly didn't vanish so much that "nobody young know what the hell it is."
If there was NO recognition, they'd have made the movie about other characters. Besides, I know a number of under 30 year olds who are well aware of the original series.
I don't know if I agree with this timeline 100%...but in the early 90's I was in my early 20's so guess it's possible that it was on and I didn't realize...I was out every night after work drinkin and partying...never home...I'm sure you all remember those days of our youth....
I remember Sci-fi getting it and they even ran marathons for awhile on various holidays and I was so happy and then poof...gone.
TV Land we can't even count -- they had it on for such a short time and they moved it all over the schedule that I couldn't even keep up.....it's now on Sunday mornings at like 5am...so really we can't count tvland..
And G4 you're right about...but I never got that channel...

I agree w/ ya that people young and old know what trek is...everybody knows kirk/spock, live long and prosper, "the enterprise", beam me up...tribbles...everybody knows it...may not like it...but they know it !