It helps to understand that Chrono Trigger's story was the result of a bit of a struggle between Yuji Horii (aka the Dragon Quest guy) and Masato Kato, who would later write and direct Chrono Cross. Horii's end was light-hearted, which makes sense given his pedigree, while Kato liked darker stuff. That's why Zeal in particular is a shift in tone from the rest of Trigger.
One of the core themes of Cross is that actions have consequences, and I personally loved how the game pulled no punches on that topic with respect to Trigger's cast. The idea of repercussions is only hinted at in the first game, but it's there, and the revelations on the beach are heavily foreshadowed within Cross's story itself. It's a grown-up narrative from an era when players were starting to demand grown-up narratives. Its reception reflects that, as well; it earned some of the highest review scores among JRPGs of its era, and it sold well enough to require reprints. This was a game that was well-received in 2000, aside from the grumblings of a few upset Trigger fans. Cross hasn't endured simply because it was very much a game of its time, and it hasn't aged as gracefully as Trigger (especially its visuals).
I consider Chrono Trigger and Chrono Cross to be an excellent cause-and-effect pairing, and Cross's connections to Trigger serve to enhance both games. I love the way Cross can reframe Trigger; I think it adds weight to the actions of a bunch of kids who stumble upon time portals and start messing with things. Time travel raises questions, and Cross's story is why I mull on those questions in my head whenever I replay Trigger.
I don't know how well it's aged for a new player, but I found it very notable at the time for being dark, if not outright macabre, at times. We had very little of that in the 16-bit era.
Drawing from real-world locales and cultures was interesting, too. Ys is another series that does that to good effect.