How do you use these character traits?

Okay, Creases, I like the "unfinished business" lens.  I'll try to keep that in mind.  The apocalypse in my game was an alien invasion that set up a xeno-terraforming process.  The humans fought in conventional ways at first and ultimately nuked the higher concentrations of alien activity all over the world, which made things much worse for humans.  Then vaguely defined weirdness happened before it rolled out into people with football pads and feathers killing each other for food and batteries.  So far we've assumed that the mission of the people in stasis is to survive and repopulate when radiation levels drop to something more reasonable, but I suppose there could be further missions.

Bignose, you're right... the archive could be useful to investigate the people in stasis.  We know that one of the people in stasis is one of the top generals who ordered the nuclear response, and have foreshadowed that he's... formidable.  Maybe finding out more about him is one way to use the archive.  Maybe the general won't be willing to let go of the past, either.  We could do that trope where the losing general never accepts that the war is over.  Assuming the quarantine wakes him up and doesn't just euthanize him in the cradle.  :)

The quarantine in my game has taken the class in a somewhat different direction, which is brilliant, but he's maybe not the typical quarantine (if that's a thing).  In some ways he's more concerned with hiding the past than revealing it.

Cool, glad I could help.

Since the quarantine's unit is military, in an alien colony situation, they might have a guerilla mission. A sleeper cell, literally.

Yeah, I guess.  Maybe my problem with it is that I don't perceive the past as being the story.  So you can find out about the plague that wiped everybody out fifty years ago... so what? 

If someone plays the Quarantine, the past starts to matter to the story. If someone plays the Quarantine and takes archives, now the past definitely matters. One would hope that this would happen naturally, simply because of the questions and moves the Quarantine playbook brings to the table -- but part of how the playbook brings that is by making the MC do a lot more thinking about the past, and about the details of the apocalypse.

This is why in general I don't suggest people just toss all the LE playbooks they own down whenever they are starting a new game, unless they're really willing to run with the choices. The difference between an AW game with a Quarantine and an AW game without one is really, really significant. Same goes for a Touchstone, and to a lesser degree some of the other books.