My cat dropped a dead bird on me when I was asleep once: “Look what I can do.” Imagine waking up to a headless bird on your chest.
My mother always got freaked out when they brought her dead birds too.
But you have to understand, to a cat a dead bird is the most wonderful gift on earth. They are showing that they really, really love you....disturbing as it may be.
Ours brought one into the house once (not dead), got clear upstairs before the bird got loose. That was not a fun time. Did something similar with a mouse a couple months ago. Fortunately, the cat cornered the mouse (this was like 24 hours later) in the bedroom, such that it went into the trap under the nightstand. He must try to keep them alive, to play with for awhile.
Now I TRY to remember to really check that the cat is coming in ALONE, no matter what time of night it is.