Bear with me, I'm making temporal omelets here...
Time eggs in the Chronoverse are quite the enigma. The only thing we know for certain about them is that they create isolated temporal distortions (Gates), and even then they seem affected by whatever the user desires most at the time. Crono's friends use one to save him before he dies. Serge uses one to reach Schala beyond time's eclipse. Kid's (in
Radical Dreamers) reacts with the Frozen Flame to reveal her true identity (meanwhile Serge witnesses apparently all of spacetime).
There's only one
1 other time when a time egg is used, and I want to focus on that: Before (after?) most of the events in
Chrono Cross, Kid, in her own relative timeline, uses a time egg to save Serge from drowning. Unique to this event, the timeline actually splits here, creating the Home world and Another with a 50/50 chance of Serge surviving. In other words, the time egg split the dimensions, but only in one of them is Kid's desire of saving Serge fulfilled. Is this event really unique however?
Consider Chronopolis for a moment, and its claim of being able to observe other dimensions.
Radical Dreamers is at least one of those dimensions, since we can read about it within Chronopolis while playing
Chrono Cross. In effect,
Radical Dreamers is absolutely canon. Not just in the usual "they're from the same series/same director, 'cept one game got retconned for another"
2 sort of way; no,
Radical Dreamers is part of the same timeline as
Chrono Trigger and
Chrono Cross, just more of an "other" timeline.
Another world. See where I'm going with this?
It's a general concensus that, in
Radical Dreamer's storyline, Crono is dead. Kid says as much by saying that Lucca wanted to bury the Frozen Flame "at her old friend's grave" so that
he could be able to rest with it. Kid even has Lucca's old time egg (though, admittedly, it could be a prototype like the one she gave Kid in
Chrono Cross). So how did Crono die? He died in 12000 BC, during the group's first encounter with Lavos, where Marle and the others, using the time egg, failed to save him. Except they also
did save him. We just have two dimensions from this point forward: One where Crono lives (
Chrono Cross), and one where Crono dies (
Radical Dreamers).
1 There's actually one other time when a time egg is used: While Kid's in her coma, Masa, Mune, and Doreen charge her prototype time egg to send Serge back in time to the burning of the orphanage. But this whole event never made much sense to begin with, since it suggests a paradox--Kid goes back in time to save Serge from drowning because Serge goes back in time to save her from burning, etc. And why was she in a coma "bound by the past" in the first place? Unless her proximity to the Frozen Flame in Chronopolis, while also carrying a time egg, triggered an event similar to what happened in Radical Dreamers... That would have been nice for the writers to touch on.
2 Though I admit, this is likely how the directors think of
Radical Dreamers. Even Masato Kato expressed embarassment with the final game.
TL;DR: Time eggs are like Schrodinger's cat, both fulfilling their users' desires while also creating an alternate dimension in which that desire is not met. Crono/Serge is both simultaneously saved/not saved, and as such,
Radical Dreamers is to
Chrono Trigger what the Home world is to Another. And if eggs = cats, and cats = Entity, then who was phone?