I looked back over the script and it looks like there might actually be two very distinct “Arbiter” statuses granted throughout the game (possibly indicating a poor translation, like how the unified Dragon God is called the Time Devourer but, to my understanding, this was not the case in the original language). To illustrate:
Only personnel registered as '"Arbiter"' may enter.
Arbiter is a status that is registered in Chronopolis. Seemingly, this is not a unique status (note: "personnel" is inherently plural).
Only personnel registered as '"Arbiter"' may enter.
Please conduct security check for verification.
Unauthorized entry or tampering with security system will result in death.
Operate at your own risk.
............
Analysis complete.
Fingerprints, retina, and C class DNA all clear. Data discrepancy due to aging within permissible range.
96% confirmation that this individual is the last registered arbiter.
Access granted.
Welcome back, Chrono Trigger.
Again, we have that Arbiter is something registered by Chronopolis' security system (and the clear indication that there was at least one, probably several, arbiter(s) before Serge). I am curious, however, if just Serge was given the label Chrono Trigger, or if every Arbiter was given that title.
Contact with the Flame healed your young body. But that was not all...
Once the security card system was rebooted, it would only grant access to you...the '"arbiter."'
And here we have two actions being performed by two different agents. On one hand, Serge is healed. On the other hand, access to the frozen flame becomes limited to the arbiter. The actors are the Flame and the security system. Given this, it seems like locking out everyone but the arbiter (and in turn, identifying who that arbiter is so it can let him in) was the result of the security system, which was quite separate from the Frozen Flame healing Serge.
It seems that it is untenable to say that the arbiter status displayed in these quotes is anything but a security rank determined by Chronopolis and Chronopolis alone.
Going back to the actual definition of the word "arbiter" itself, this is probably specifically a status given to researchers (those who were testing and investigating the flame and "judging" their results).
However, "arbiter" is mentioned one other time in Chrono Cross:
A new species is about to be born on this planet -- an alien life-form even more evolved than the old Lavos!
At the darkness beyond time, the weakened Schala came under the influence of Lavos, and the two became one entity.
It is now up to you, the one whom the Frozen Flame has chosen as its '"arbiter"'...
You alone can decide how the new Lavos, which has encaged Schala within it, will evolve from here!
Your actions will determine whether in the future all time is devoured by Lavos, sending the world into everlasting death.
Belthasar foresaw this was going to happen, in his world in the year 2300.
And he was determined to prevent it from happening, no matter what it took...
Actually, this quote has a lot of meat to it, but let me start with "Arbiter."
Here the Frozen Flame quite literally chooses Serge as the Arbiter, yet notably this is expressed by Marle, not Chronopolis. This being the (maybe) same Marle who (along with Crono and Lucca) accused Serge of "murdering" the future in a rather poetic manner. It is tenuous at best to link this status back to the status seen in Chronopolis, especially given what follows.
Marle tells us exactly what Serge is "judging" (remember, the definition of Arbiter is one with the authority to judge). As having
this status of Arbiter, Serge is essentially judging how a new life form, the Time Devourer, gets to evolve (note, it is curious that Marle says that Serge is only determining how the TD will evolve, not if it will live or die). Schala, Zeal, FATE, even Belthasar were not the ones to have that sort of authority; they cannot have held the same Arbiter status that Marle is here ascribing to Serge.
The Frozen Flame is what chose Serge as the arbiter... but the Frozen Flame is a representation of Lavos, so it would seem that Lavos then was the one to choose Serge as the arbiter (and why give a human that authority?). However, Schala is the one encaged in Lavos (... almost like Schala is replacing the Lavos Core and bits... yay for circumstantial evidence that supports my multi-organism life-form definition of Lavos), so it may be her influence on the entirety of the creature that is actually what is communicating, through the FF, that Serge has authority.
Regardless, neither variations of Arbiter status, from the game, can apply to Schala or Zeal. However, one who touches the flame may gain a special connection (indeed, it is that connection that supposedly evolved humans), but this is not the same as Arbiter status (and in turn, Schala wasn't the arbiter of the frozen flame and Zeal wasn't the arbiter of Lavos). We might want to use the world Avatar instead, to make this distinction clear.
For some of the other meat of that quote, this actually seems to confirm that the TD doesn't consume time-space like a sandwich. Admittedly, when I first encountered the statement that he essentially ate time itself, I was imagining a sort of nonexistence left in its wake; it ate time-space, so time-space wasn't there anymore. But Marle clearly establishes that something still exists after the TD devours it... that something just happens to also be dead, and irrevocably so. Sort of like the Ruined Future, but for all of Space-Time.
To summarize, the TD does not "consume" space-time anymore than Lavos "consumed" the planet. Lavos was feeding off the energy, not the magma, precious metals, and soil of the planet. Likewise, the TD seems like it would have feed off the energy of space-time, not the specific locations and timelines themselves. We might equate the end result of the TD as being like heat-death, only cold-death. There is no energy so nothing can ever happen.
Also, it is quite clearly established that the TD was to be born there on Opassa Beach. This seems to make it clear that the Time Devourer would have been able to use the gate Serge created to return to real time, or at least have a connection to real time.
Finally, that quote seems to indicate that Serge did not destroy the Time Devourer. Rather, he changed its evolution but it is implied that it still exists. Perhaps it merely reverted to Lavos and was lost to the DBT, perhaps it reverted to Lavos and awaits for a new victim to find its way into the DBT (look out Janus). Or perhaps Schala might even be the result of this "evolution." I think Kato may have been specifically giving himself room for a third installment.
