But why would Savathûn or Quria work with the Nine?

They aren't, from my perspective it's more like the 9 are trying to understand beings that have the ability to change their very own nature (fate). All hive are destined to die by their worms if they don't feed it with chaos, savathun tricked it by having quaria create the dreaming city curse cycle and have guardians partake in an endless war against the taken, in doing so she has not only prolonged her life but has also found a new way to conquer her own fate. Quaria on the other hand is capable of simulating entire realities into existence. While doing savathun's bidding it is also trying to find a way to free itself from her grip. And now the drifter, a man who is a being of light but doesn't call himself a guardian, who uses darkenss & it's powers but doesn't side with the shadows of yor. In a way he is using both the sides but stays neutral. The 9 are intrigued by this. The new lore pieces under the 'dust' lore section tell us that the 9 are the corporeal form of dark matter with a consious. they need us to survive for them to exist. But they also want to break this dependence they have on us and hence look to those who have done it before.