I think their victory in the war gives the White Council a legitimate claim on the territory. That is not nothing, but it is not enough to keep the territory by itself. If the White Council lacks the strength or will to defend the territory, other supernatural races WILL move in and take over the territory. The supernatural races are almost all predators after all.
I think if conditions were different, the White Council could probably claim and keep a big chunk of the captured territory, but even under ideal conditions the White Council would have lacked the numbers/strength to hold ALL the former Red Court's territory. After all, it would be a doubling of the size of White Council territory (which is mostly Europe and North America prior to the war) and it is not exactly easy to govern either (dysfunctional governments, criminal gangs, power vacuums, income disparities, ect). That is an awful lot to absorb and defend even if conditions were ideal. And things are certainly NOT ideal now.