The only problem I see with that is one of the Compendium's staple theories, Time Traveler's Immunity. If Kid traveled 10 years back in time, from that point she would be protected from any changes in the timeline, including her own.
The Kid we interact with in the game is pre-time travel Kid. She doesn't know about all this stuff until the infodump at Opassa Beach.
My problem with the whole thing is predestination, which Chrono Trigger was definitely against. Kid is going to save Serge someday, so Serge is saved. But the future doesn't change because you're going to do something, it only changes when you put the effort in. Otherwise CT would have ended with Crono watching the recording of the Day of Lavos; now that he knows about Lavos, he's going to stop it, so Lavos is stopped.
The fact that we know Kid comes back from the future, means there is a future for her to come back from, which means the Time Devourer failed to destroy all space-time. So our victory against it is foreordained and we're just going through the motions. It robs the story of some of its impact. Maybe I'm just overthinking it, but I really wish the writers had come up with some other explanation for Serge's survival.
I mean, it's never specified when the TD reaches maturity and what exactly happens then. With Lavos there was a clear date on the apocalypse, it always happened in 1999 AD. It's said that the TD will devour all spacetime, which I take as all time and space everywhere ceases to exist. So if we assume the TD being succesful in doing so is the default, doesn't that mean that all spacetime should cease to exist the moment he starts bonding with Schala? But that's also predestination.
The same thing can be extended to Chrono Trigger. Lavos arrives on Earth, so the world is destroyed.
Time flows in a linear fashion in Chrono Trigger and Cross. You can go forward or backward, but events have to play out before they happen. There is a predestination in place but because time flows the way it does, that predestination can be altered if someone goes back in time and changes the events that lead to it. The argument that Kid would need a future to return to after saving Serge does not make much sense to me for example, because Chronopolis Time Crash happened in 2400 AD. No matter the outcome, she always had a future to return to.
The fact that we do not know exactly when the TD matures and devours everything does not matter due to this. Everything before that time would still happen in a linear fashion. Lets say, after Lavos is defeated, Crono and gang jump on the epoch and travel to...3000 AD and TD matures in 2900 AD. They would arrive in a void of nothingness. No dimensions, no "End of Time" platform...just...blank.
But if they went back in time, reality would still exist during the time they went back to. I know Bathasar SAYS all of spcae time, but it is not that simple. The entire premise would become a mess if it was not linear. There would have been no deleted future, as Crono would have been predestined to kill Lavos...which means that from the moment Lavos lands on earth, the TD would have already been created, which means that had Serge failed, both games would have never happened. Then there is the bad ending, which was stated in Ultimania that if you kill TD without the Chrono Cross, you merge with it and all of spacetime in consumed. This is represented by there ne NO ending at all. Everything still happens up to that point though.
So the only way all of this really works, is that time flows in a linear fashion. This is seen in both games, and all evidence points to it. Until a certain event happens, there is a destination for said event. If someone alters time and changes that destination, that old destination is erased and gets sent into the DBT. If you travel to the future before the destination is changed, you will see it play out as intended.