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Gandalf was not supposed to care about the east
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Actually he was and did, he just thought that the best resistance to Sauron would come from the west, primarily the remaining elves, the descendents of the edain and their allies such as Rohan.
Denethor was entirely focused on Gondor and therefore his own power. He thought very little of the other inhabitants of Middle Earth including Gondor's allies. He thought of Sauron as a political rival, as someone who should be defeated because he infringed on Gondor's power not someone who should be defeated because they are a tyrant and are evil.