It seems clear to me, that the missing link of the CC story has to be Chrono. Something has to account for chrono's absence, we all know he'd be there, kicking ass if he could. There's alot going on with the frozen flame that's unclear, then there's the trouble with reconciling a hero in this world(Serge) that isn't Chrono to begin with. Both are silent, ambiguous characters, so as a player it's unfair to leave us with nothing to distinguish the two (serge and chrono) from each other.
I say missing link, because it's blatantly obvious that Square ultimately did not put 100 percent effort in the sequel. The scope, and feel of CT is so much bigger than CC, with it's small map, and repetitive areas. You can try to explain this discrepancy with the plot device of Fate, sure... but it's still cheating us from the rest of the game. No surprises really though, we all had a hunch when we bought it and noticed two cd's... when final fantasy programming was taking up 3 and 4 discs a game. Even when you account for FMV, there's still so much game missing. Another disc or 2 would take us out of the ridiculously small island, to go see Guardia, etc. And heaven forbid they let us do any time traveling in a Chrono game...
So yeah, it's frutstrating, and we're all left guessing as to what was missing amidst ridiculous claims by the game designers that their wasn't much missing at all. My guess is that Serge's existence was some sort of contradiction to Chrono's existence, them both not being able to exist at same time, in same world, dimensions, etc. Maybe they're the same person, maybe not.. doesn't matter. I won't guess as to the specifics, because Square likes to make those as confusing as possible, but it would go something to the effect of....
frozen flame, blah blah, serge born blah, blah, 5 horribly confusing cutscene's blah blah... and then you say "WOW, that's why they called him the assassin of time." I can't think of any other explanation that would reconcile the strong characters of both games.
Also, I like this because it's a play on words.... assassin of 'time', or otherwise translated as assassin of 'chrono.'