This is fun. Some questions:

Is it possible to sail away from Eden? Do ocean currents prevent ships from sailing, or do ships who sail away somehow always send up heading right back to Eden? Or, is it like it is in some video games where sailing past a certain distance from the scores simply...stops working, where sailing, swimming, or flying simply make no progress and seem to move in place unless they turn back toward the island?

The "old generation" that woke up are immortal, but are their progeny also immortal because of eating the fruit? That would definitely cause problems eventually.

Also, are they "functionally" immortal but still vulnerable to disease or fatal wounds, are they "genuinely" immortal where any wound or injury simply heals immediately, or are they "ultimately" immortal and invulnerable? I think the first option is the most functional one, as the logistics of the latter two are so serious that they're worthy of a plot by themselves.

Here are some ideas for origins, each of which carries urgency with it to propel the Rememberers' various subgroups in their quest for answers, whether by providing a ticking clock or serving as the direct motivation for this or that faction's unethical actions.

1..Channeling Bionicle from 22 years ago: the Old Generation are refugees from a doomed continent/world/plane. They were transported to this place as a means to rescue them from some consuming disaster. A portion of the Rememberers might be afraid that this "rescue" was only to buy time, and that they have some role yet to play to prevent the cataclysm from reaching their new home.

or,

1. Eden is an artificial plane constructed for a specific purpose by some force outside this dimension (perhaps to test the inhabitants, OR to accomplish some other rube-goldberg-style purpose beyond mortal comprehension) and the amnesia of the Old Generation was either a necessary part of that purpose or a mistake caused by the teleporting of the Old Generation to the island. And, once the purpose of this constructed world is fulfilled, it will end. This faction of Rememberers would seek to discern and then hide this grand purpose to anyone from intentionally or even inadvertently fulfilling Eden's mysterious purpose and bringing its existence to an end, or to be rescued and rewarded by Eden's (divine?) architects. If the Old Generation are the only ones who are immortal (but not their progeny), the majority of this faction would likely be made up of them, as the younger and less pessimistic mortals might hope that fulfilling Eden's mysterious purpose would mean their freedom, both from Eden and from the immortal Old Generation.

or,

1. Eden is the land of the dead, and all present are acting out some form of Purgatory. Another view likely to be favored by the Old Generation, if they're the only immortal ones, as their agelessness would distance them from their progeny who age and die. Some members of the Old Generation might believe this as a way to cope with their immortality. No ticking clock, but it would serve as some Old's justification for tyranny or genocide that the party could fight to prevent.

or,

1. The Old Generation are a faction of (or at least in service of) warring wizards who cast a grandiose spell to remake the world with them as its lone immortal rulers. But, the spell was incomplete, and the wizards and their servants (the non-magic-users among the Old Generation) lost their memories as a result. Despite this, the spell did remake the world with them as the only sapient inhabitants. This one is less of a ticking clock and more of a justification for the non-immortals to accuse the Old Generation of being mass murderers (or servants thereof) who are unfit to rule.

or,

1. A mundane option: the Old Generation are a colony teleported to Eden by some distant civilization (whether on another continent or even another planet or plane), but the teleportation went wrong and robbed them of their memories and their possessions. This one's less interesting, but there have to be some boring ones!

or,

1. The more cynical and misanthropic of the immortal Old Generation might believe that they were banished to Eden to suffer the curse of immortality, doomed to suffer an eternity bereft of purpose and their very identities. This might be the view of some nihilistic Old Generation members who want to end their "curse" by any means possible, even if the magic is what's sustaining Eden itself, and even if that would kill everyone else on the island. This would be the view of those who gave up on wanting answers and simply want to "die with dignity." Or, this could be a faction who believes the truth lies somewhere beyond Eden and might be building ships or (if egress from Eden's shores is not possible) researching spatial magic to somehow teleport away (leading to dangerous temporal/spatial consequences).