JKR also said that you have to dwell on it a bit, to really understand the finality and reality of death.
That's why Harry didn't even see the Thestrals on the way back to the Hogwarts Express at the end of GoF. He was still in shock, and a bit in denial.
You may have heard of the 7 (some say just 5 )stages of grief: Shock, Denial, Grief, Guilt, Anger, Depression, and Acceptance:

You don't see the thestrals until you go through all of them, and though they sometimes skip or jump, it always ends with acceptance.