I have played quite a bit more and I would like to share my updated findings.
Having done some comparissons myself, I've found that the remastered version runs worse than the original and emulated versions. While Chrono Cross did have framerate drops in the original release, they are much more severe in the remastered one. Framerate drops are much more common in towns and the frametime is also all over the place.
This becomes an issue in battle too. The framerate tanks during combat in both the original and the remastered version, but the remastered version comes with another annoying issue: Huge input delay, which makes selecting commands a chore. Sometimes, your commands won't even be registered at all which is quite annoying.
Now the classic mode is....Fine. This one actually runs pretty much identical to the original, but with a higher resolution and added Anti Aliasing. It doesn't seem to have the input delay during combat or at least not as noticeable.
I still prefer the remastered version because the new character models and portraits look really good, but there's no denying that this is an incredibly dissapointing "port" and it's basically just the game running on an emulator with enhancements.