I can easily see where some stories simply wouldn't work for newer audiences, even though I love them. One that I would point to right off the bat is The Conscience of the King
(an episode I love). My daughter asked a very straight forward question when she was a bit younger: why don't they simply do a DNA test? I know and you know that wasn't even on the radar in 1960's storytelling. But I could easily see it being a stumbling block to someone who has known about DNA almost their entire lives.
And this of course is where you explain that DNA testing for criminal investigations wasn't even on the radar in the 1960's.
I think it is hard for a lot of people (not just kids) to accept that Sci-fi about the future doesn't show you the future but a reflection of the present of the time period it was made.
I think to help kids appreciate something classic you have to give them a little historical context. And you can say while they didn't predict the DNA thing here are a few things that the series did predict.