Much as I liked the tenacity and resourcefulness of Nate's character in "The Birds" (1951), the chances of him and his family making it alive out of the apocalypse in the story seem bleak. Although Daphne Du Maurier leaves the ending open, this only serves to add to...
the mood of bleak uncertainty that has been growing in the story, leaving us pessimistic about the little family's fate. But apart from the conventions of the horrorgenre, which generally demand a tragic ending, there are three other reasons I think Nate and his family will not survive the prevailing conditions in "The Birds."
One, as you rightly noted, is the foreshadowing of human extinction in the narrative. By day 3 of the bird-attack, the Triggs have met a ghastly end as has the postman and also, it is suggested, the families living in the "council flats." The birds have crashed rescue airplanes and even the wireless has gone silent. As humans fade, the birds have only been increasing in number and power. They also seem to be evolving rapidly in intelligence, suggesting they will outmaneuver Nate soon:
The smaller birds were at the window now. He recognized the light tap-tapping of their beaks and the soft brush of their wings. The hawks ignored the windows. They concentrated their attack upon the door. Nat listened to the tearing sound of splintering wood and wondered how many million years of memory were stored in those little brains, behind the stabbing beaks, the piercing eyes, now giving them this instinct to destroy mankind with all the deft precision of machines.
Two is the story's implicit theme of the dangers of climate change. Time and again people in the story postulate that the change in the behavior of the birds is linked with "change" in the "arctic circle." Something has happened which has changed nature as represented by the birds from a nurturing entity to a dark machine of destruction. With humans at the receiving end of avian fury, Du Maurier is suggesting the crisis may have been propelled by human actions. It is significant that today, many critics read "The Birds" as one of the early prescient stories about the impact of climate change. The lesson of the story thus can be fully delivered only with the extinction of all humans, which sadly includes Nate and his family.
Lastly, "The Birds" is also an allegory for some of the events of World War II. Like the Triggs initially ignore Nate's warnings about the birds, people over Europe too were blind to the rise of fascism. The aerial attack of the birds recalls the air-raids over Britain during the war, and Nate's wife's query recalls America's role in helping end World War II:
Won't America do something….They've always been our allies, haven't they? Surely America will do something?
The reference to war and the continuing ignorance of everyone other than Nate point to a world which has not learnt much from its mistakes. Therefore, doomsday is at hand for the inhabitants of this world.