I think that's a fair summary, Samt.
By the time we encounter the Aes Sedai, it appears that the main purpose of the White Tower is to consolidate/maintain the power of the White Tower. A few people in it, i.e. Moiraine and Siuan and Nynaeve, are working towards the greater good (the greaaater gooood) but most Aes Sedai are pretty Machiavellian about gaining power and influence for the sake of power and influence.
Therefore, some of the 'missions' as you put it, of the Ajahs could be vestigial remains from the Age of Legends. That was a technologically advanced society, more akin to the 21st century, but knocked back to the 16th century (give or take) after the Breaking. The AoL Yellows could have been pursuing new knowledge in medicine, and the AoL Whites could have been using math and logic to advance science and technology. AoL Greens were probably not useless and doing more than having Warder orgies.
Browns were probably like people who have PhDs in history or philosophy. Reading lots of books and forever wondering why they joined that Ajah because there are no jobs and no one gives a damn (yes, I have a PhD in history).
Joking aside, by the time we meet these people in the Third Age, the Tower has stagnated, and they are doing none of these things. Just being catty and manipulative, maintaining a status quo that attempts to preserve the White Tower as the most geopolitically significant power in Randland, but not actually using the One Power or the vast knowledge contained in their library to advance society.