Despair? I thought JKR mentioned depression, not despair. And the dementors are grey, not green.

And JKR's own experience with depression was after the death of her mother -- so the "death causes depression" is similar in content to "Dementor destroys all positive emotions". There's no reason why JKR couldn't use imagery of death in combination with her emotion.

In the books, an afterlife exists. But the soul-destruction, the true death, is what the Dementor seems to cause. Again -- if JKR's mother had just died, her most depressed thoughts about it would be the ones that involved thinking about the true death of soul-annihilation (the Dementor's kiss), not an afterlife.