Given the tradition of the Chrono Series, original characters seem almost more canonical than fanfic about established ones. This does push the sort into the realm of "might be better off as an independent franchise," but personally I find the best aspects of the game to be the implications for adding layers of complexity (something that could be done with an independent franchise, but not as easily). Like everyone and their mother, I've worked on my own fan version of a sequel: it is specifically built on what came before in CT and CC. The cast of those games barely makes an appearance, so it would be easy to cut it loose in that regard, but then to really tell the same story I'd have to recreate CT and CC.
So I guess, if one wants to just make a fan fic about Chrono Trigger or Chrono Cross, then original characters aren't necessary. But if one wants to make a fanfic about the Chrono Series, as a continuation of it, then original characters seem to be fundamentally necessary.