Cool is an elastic thing. Sometimes it means popular. Sometimes it means wonderfully quirky. (I’m thinking of 11th Doctor’s descriptions of his unusual sartorial choices.)
Kids like to fit in with their peers, so cool to them probably means what everyone else is into. I didn’t like lounge music when I was a kid because it was of my parents’ generation. I didn’t like opera when I was a kid because I had little exposure to it, so it was weird. Happily, my tastes have broadened. I regret it didn’t happen sooner.
Star Trek has become classic (except, maybe, Enterprise) for good reason. That’s enough. They snagged James Darren at the end of DS9. Great. Some kids didn’t get Vic Fontaine at first blush? Too bad. Try again. The best art requires a little effort.
Also, when you love something, your enthusiasm pulls in others. If they reject it because it’s not popular, fuck 'em. Wonderfully quirky people are so much more interesting than people who fit in.
In fairness, though, Patrick Stewart did Shakespeare before he did Picard. Have you seen him as a young man playing Oberon? https://www.google.com/search?q=patrick+stewart+oberon&tbm=isch&sa=X&ved=2ahUKEwjStue03pOAAxU6XUEAHb4ZDLUQ0pQJegQICxAB&biw=1685&bih=803&dpr=1.14#imgrc=6p-8DlObJa4hpM