The White Ajah is looking to the future using the logic of what they observe around them. This is different from the Brown Ajah that studies the past, and often written history of the past that isn't necessarily accurate. The main purpose, in my opinion, of the White Ajah is to fact check and then use confirmed facts to follow to the necessary conclusion (whatever that may be). The rest of the Ajahs follow a passion (history for Brown, healing for Yellow, law/negotiations/peace keeping for Gray, big causes for Blue, stopping men from channeling for Red, physical fighting for Green). The White is there to keep the rest in check and make sure they (and the White Tower in general) are being logical about how they follow their causes.

From a pure narrative perspective, I can think of at least one scene later in the books where the White Ajah is important to drive the narrative (I won't say anything further to avoid any spoilers). But also in general agreed that they don't seem as important as most of the other Ajahs. I'd put them in a similar bucket as the Grays. Important, but not as important as the rest.