I will note that this is not correct.
A lot of Greens spend their time not in the Borderlands, but traveling the land meting out justice to criminals who are beyond the reach of the local justice system - escaped murderers and the like. This usually involves their Warders' swords.
Brown concern themselves with knowledge, and certainly do not stay cooped up in libraries. Some might, but many are archaeologists, anthropologists, geographers, botanists, ornithologists, and so on so forth for a very long time.
As for Reds, we do see a handful of sisters who are reasonable - Pevara, Silviana, Teslyn.
The Yellow Ajah are not in focus in the novels, since the only prominent member we see is Nynaeve and her activities are rather irregular. However, for the purposes of this quest assume that the Yellow Ajah, in addition to maintaining the Tower's legendary infirmary that turns away none seeking help, also travel far and wide to cure those too weak to seek help in Tar Valon.
Couple of further highlights:
-Greens are as mentioned, the Battle Ajah, but the lack of wars they are permitted to participate in due to the Oaths means that a fair number of them come from the Borderlands and return home to the Borderlands to help against Shadowspawn. As we've seen with Alanna and Cadsuane, they also do the adventurer thing with their
harem Warders. Show up, save the day, ride off into the sunset. Worth noting that they are very numerous, because the Borderlands take their arrangement to send prospective channelers to the Aes Sedai very seriously.
-Blues we see a lot in canon, and they're the Intrigue Idealists of the Tower. Every Blue of mention has extensive spy networks mentioned, and they're the most adept at working around the Oaths to do strange fishy things, or even personally infiltrate places..
-Greys are the stereotypical Courtly Aes Sedai. Unfortunately for them, the events of canon really put them in a pinch, as social order breaks down while they frantically try to save the status quo, but below the level of nobility they are important arbitrators and negotiators throughout the land.
-Browns are clustered around power centers mostly. With the communications and scribing technology of the era, they're less cooped up by choice and more that they are
intensely occupied trying to identify and rescue lost works, often buried in some noble's vanity collection and never read. We see a lot of the more nerdy types staying satisfied in the Tower, but for those out with a Mission, diplomatic chops are crucial to sweettalking a proud noble to let you see and copy from his collection. They don't really have the TIME or manpower to salvage all that they would prefer.
-The Whites are the actual scientists here, though they call it philosophy, they spend their time finding new proofs, though what you get left in the Tower, again, are mainly the mathematicians and theoretical physicists of the lot. The rest are out there doing naturalist studies and gathering data for their theories, because...well, again post apocalyptic, if you want your proofs, find them in person.
-The Yellows, from what little we know of them, are simply too few to actually do things hands on, though no small number would make it their lifes work to do the wandering healer thing. Their biggest role is to identify and halt pestilence in it's tracks, which is a PRETTY intensive labor, because once a plague goes out of control the whole Tower would be hard pressed to stop it. But if they can get there in time, a small group of Yellow healers could save thousands of lives by healing a few dozen people.
-The Reds, despite their showing in canon have their charter to be
combating abuses of the Power, while true that the narrative centered around the management of channeling men, they are the ones who hunt darkfriend Aes Sedai, those who abuse the Power to practice forbidden weaves like Compulsion and Balefire, deal with those Aes Sedai overstepping their bounds quietly, etc.
Is there any surprise then, that the Black Ajah made it a mission of theirs to take control of the Reds over the others? We know none of their power structures are immune. But they made it a major objective to subvert the Reds, because the Reds are the ones tasked to eliminate the Black in their founding charter.
