Hi there!
First time poster, long time lurker here.
Every now and then, I get captured in Chrono series lore and this last time, I caught myself thinking about something similar to this question.
"Why only Kid's time traveling ended up in dimension splitting?"
As I couldn't come up with an answer (or find one), something hit me: "Who said other time travels didn't split dimension either?"
It's probably a long shot but, I kinda came up with this:
Time travelling and changes in history result in dimension splitting. The "active" (or Keystone) timeline is rewritten and the future which was changed, exists in another dimension that split from Keystone and end up in the Darkness Beyond Time (like the little glowing orbs that float around when you battle the Time Devourer).
Now, ending up in DBT doesn't mean that dimension/timeline is erased. It continues to exist. One argument supporting it is Dinopolis, which was pulled from a dimension/timeline that was discarded once Lavos crashed into the planet. It actually came to exist in that dimension way after the day is was split from Keystone (when Lavos crashed).
I guess my point is: There's always a rewrite and a split. Keystone is rewritten, discarded futures are results from splits.
Dwelling a bit further, when Kid changes history, there is a rewrite (Home) and a split (Another). Serge was supposed to die in Keystone. Kid rewrites it saving him. The future where Serge is dead is discarded to DBT. It still exists, like the Reptite timeline/dimension, hence it can be reached through the distortion at Opassa Beach.
Now, if that's the case, what's the difference between Keystone and other timelines/dimensions?
Not sure. But maybe, the only way to physically reach the DBT as a place (and thus, kill the Time Devourer and free Schala) is through the Keystone timeline. Thus the need to have Serge alive in Keystone, so he can execute Balthazar's plan.
Bonus crazy theory: Based on this principle, we could answer the question: "How does Serge being alive spells destruction for the future?"
It doesn't. Crono's actions which ensured that the Day of Lavos didn't happen, occured before the rewrite done by Kid. Thus, it's not affected (as, rewrites would change history only from that point after).
"What about the Tower of Geddon and the Dead Sea?"
It's not a reflection of a future destroyed during the Day of Lavos. It's a reflection of the destruction that will be wrought by the Time Devourer (the "frozen in time" aspect of the place showing exactly that. Time doesn't exist there anymore. It was consumed.)
The ghost children blaming Serge would be ploys by FATE, trying to make Serge kill himself (and release the lock on the Frozen Flame), if he felt guilty enough, feeling responsible for the destruction of the future.
No idea how bananas (or dumb) all of this sounds but, I really love the work on this site and I felt I could chip in with some thoughts.